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JPL PUBLICATION 80-77, Volume I

(NASA-Cl3-163826) CONTROL ANL LYNAMICS STUDY NBI-14395FOR THE SATELL11E POWER SYSTEM. VOLUME 1:MPTS/SPS COLLECTOR DYNAMIC ANALYSIS ANDSURFACE DEFORMATIUN (Jet PrOVUISion Lah.) Unclai;113 p 11C A06/MF A01 CSCI 10A CJ/4# 29.558

Control and Dynamics Study for theSatellite Power .SVolume 1. MPTS/SPS Collector Dynamic Andjysisand Surface DeformationS. J. Wang

D5

10

"

September 1, 1980

National Aeronautics andSpace-Administration

Jet Propulsion LaboratoryCalifornia Institute' of TechnologyPasadena, California

JPL PUBLICATION 80.77, Volume I

N

Control and Dynamics Study for thed

Satellite Power SystemVolume I. MPTS/SPS Collector Dynamic Analysisand Surface Deformation

S. J. Wang

A

September 1, 1980M

National Aeronautics andSpace Administration

Jet Propulsion LaboratoryCalifornia Institute of TechnologyPasadena, California

The research desci:bed in this publication was carried out by the Jet PropulsionLaboratory, California Institute of Technology, under NASA Contract No. NAS7-100.

j

Acknowledgements

The author wishes to express his appreciation to R.C. Chernoff,

AA R.M. Dickinson, J.N. Juang, S.L. Miller and G. Rodriguez of JPL, to

R.C. Ried t and F.J. Stebbins of NASA JSC, and to S.V. Manson of NASA

Headquarters for their valuable contributions and Assistance.

w

A

a

M

i

L

Abstract

This report presents the results of part l of the dynamics

and control analysis for the MPTS antenna and collector interactions.

The objectives of this part of the study have been to establish the

basic dynamic properties and performance characteristics of the antenna

so that the results can be used for developing criteria, regvirements,

and constraints for the control and structure design. Specifically,

the vibrational properties, the surface deformation, and the corresponding

scan loss under the influence of disturbances have been studied.

TABLE OF CONTENTS

!mse

List of Illustrations, . . . . . . . , . vi

List of Tables . . . . . . N vii

1. INTRODUCTION AND SUMMARY . . . . . . . . . . I

161 GENERAL BACKGROUND . . . .

1.2 PERFORMANCE REQUIREMENT GUIDELINE . . . . . . . . . . . . . 2

1,3 OBJECTIVES AND SCOPE . . . . 2

1.4 STUDY RESULTS AND CONCLUSIONS . . . . . . 3

2. DYNAMIC MODEL AND PERFORMANCE RELATED EQUATIONS . . . . . . . . . 8

2.1 THE DYNAMIC EQUATIONS AND ENERGY OF CONCENTRATED MASSES . . 8

2.2 LOCAL SLOPES OF THE SECONDARY STRUCTURE . . , . . . . . . . 11

2.3 THE MICROWAVE TRANSMISSION SYSTEM SCAN LOSSES . . . . . . . 20

3. SURFACE DEFORMATION AND ITS EFFECTS ON THE ANTENNA EFFICIENCYDUE TO DISTURBANCES AND DYNAMIC INTERACTIONS . . . . . . . . . . 26

4

3.1 THE MODAL CHARACTERISTICS OF THE ANTENNA AND THEIR EFFECTSON THE OVERALL STRUCTURE OF SPS . . . . i . . . . . . . . . 26

3.2 ESTIMATES OF DISTURBANCES AND DYNAMIC INTERACTIONS . . . . . 30

j 3.3t

SIMULATION, RESULTS, AND DISCUSSION . . . . . . . . . . . . 38

3.3.1 Types of Forcing Functions . . . . . . . . . . . . . 38

3.3.2 Simulation Time Considerations . . . . . 41

3.3.3 The Simulation Outputs . . . . . . 41

3.3.4 The Surface Deformation and the Scan Loss . . . . . . 42

A. Surface Flatness Subject to Collector BendingOscillation . . . . . . . . . . . . . . . . . . 42

B. Surface Response to Collector Bending Disturbancedue to Thermal. Distortion . . . . . . . 44

C. Surface Deformation Subject to CollectorTorsional Oscillation . . . . . . . . . . . . . 46

D. Comparative Performance Analysis of AntennaUnder Rectangular-Pulse-Forced Acceleration . . . 48

3.3.5 Further Discussion , . . . . . 0 . . . . . . . . 49

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . 93

APPENDIX A --Local Slope and Scan Loss Program . . . . . . . . . , 94

APPENDIX B -- 3-D Plotting Program of Spatial Surfaces . . . . . . 102

v

List of Illustrations

!Me

1. The MPTS/SPS System Configuration . . . . . . . ► . . . . 12

2. Grid Points of the Finite Element Model and theSecondary Structure Supporting Pins . 0 . . . . . . . . . . . 16

3. The Grid Points and the 61 Hexagonal Planes . . . . . . . . . 17

4. Plane Intercepts . . . . . . . . . . . . . . . . . . . . . 17

5. Array Pattern and Element 'it^:ern for RectangularAperture Array with Uniform Illumination . . . . . . . . . . . 22

6. Beam Amplitude vs, Scan Angle . . . . . . . . . . . . . . . . 22

7. Scan Toss of a Rigid Array . . . . . . . . . . . . • . • . . . 24

8. Mode Shapes of the First 14 Flexible Modes ofthe MPTS Antenna, . . . . . . . . . . . . . . . . . . . . 28

9. Modal Frequencies of the MPTS and SPS . . . . . . . .6 32

10. Gravity Gradient at Three Orbit Positions of theMPTS/SPS Oscillating with a 24-hourPeriod . . . . . . . . . . . . . . . . . . . . . . . 32

11. Collector Bending due to Thermal Expansion . . . . . . . . . . 34

12. The First Torsional Mode of the Collector . . . . . . . . . . 39

13. Positions of the Applied Forces . . . . . . . . . . . . . . 40

14. Surface Response to Collector Bending Oscillation;zm = 10m, F - 6832.8 cos (.006751) . . . . . . . . . . . . 52

i

I15. Surface Response to Collector Bending Oscillation;

. Lm n 10m, F = 6832.8 sin (.00675t) . . . . . . . . . . . . 58

16. Surface Response to Step Acceleration, F = 6832.8 U(t) . . . . 64

17. Surface Response to Collector Thermal Bending.Disturbance, a = 10- 6 m/m/°C, F r 4558.5 cos (.006751) . 66

18. Surface Response to Collector Thermal BendingDisturbance, a - .75 x 1041nie i/° C, F = 3417. 2 cos (.00675t) 68

i. 19. Surface Response to Collector Torsional Oscillation,j 0 = 1 Q 9 T = 6.453 x 106cos (.02261) . . . . . . . . . • 70

r

ì

i

Page

20, Surface Response to Collector Torsional Oscillation,am * 1.,?° a T • 8.327 x 10 sin (.0226t) . . . . . . . , . , 76

21, Surface Response to Rectangular Pulse Acceleration -Translation, F . 4000 (U(t)-U(t-500)) . . . . . . . . , 7$

22. Surface Response to Rectangular Pulse Acceleration -Rotation, T - 2 x 106 (U(t)-U(t-500)} . . . . . . . . . . . . 84

^ y

23. MPTS Surface Deformation vs. Amplitude of CollectorBending Oscillation , . , . . . , . , 0 . , . . . . . , , . 90

24. MPTS Surface Deformation vs. Amplitude of CollectorTorsional Oscillation . . . . , . . . . . . . . . . . . . . . 91

25. Collector/Antenna Boundary Displacement vs.Coefficient of Thermal Expansion of Solar CollectorStructure . . . . , . . . .0 . . . . , . . . , . . 92

4

R

vii

List of Tables

pA^

1. The First 20 Modal Frequencies of the Antenna Structure . 27

2. Bonding Displacement . , . . . , . . . . . . . . . . . . 36

3. Thermal Disturbance Estimates . . . . . . . , , , , 36

4. Comparative Performance Analysis, Z ^► 10 m . . , , , . 45

5. Analysis of Performance: Distortions Caused byThermal Disturbance, a w 10-6 and .75 x 10-6 m/m/°C . . . 45

6 Analysis of Performance: Surface Deformation Causedby Collector Torsional Vibration . . . . . . . . . . . . . . . . 47

7. Comparative Performance Analysis of Antenna underRectangular-Pulse--Forced Acceleration . . . . . u . . . . . 47

8. Key to Variable Names for Figures 14 to 22 . . . . . . . . . 51

viii

F

^R

SECTION l

INTRODUCTION AND SUMMARY

This report presents the results of Part I of a two-part studyof the dynamics and control analysis of the MPTS * Antenna and its inter-

action with the solar collector. The effort reported here deals with the

dynamics problem of the antenna/collector, especially the surface deforma-

tion and the effects of the power loss due to the warping of the antenna

surface. The second part of this work deals with the attitude and pointingcontrol problems which will be summarized in Volume II of this report.Although the main emphasis of this report is on the MPTS antenna, the over-

all results of this study apply to the SPS configuration formed by a

collector structure plus two end-mounted antennas (see Fig. 1, in Section 2.2).

1.1 GENERAL BACKGROUND

The SPS is the Largest space s ystem conceived to date that

appears feasible with reasonable extensions of existing control. technology.

It represents a class of large platform-Like structures that are several.

orders of magnitude larger than any of the other large space systems

(multiple-payload platforms, parabolic reflectors, etc.) currently in

planning within NASA. The SPS has in common with all large space systems

many control problems that are widely recognized within the controls

community. These problems include attitude errors due to disturbances,

potential instabilities due to truncated modes and other model errors, lank

of damping, inaccurate preflight knowledge of the vehicle dynamics, and the

parameter variations while the system is in operation. The qualitative

nature of these problems (model errors, concentrated stresses due to Large

actuator size, etc.) has emerged as a result of studies in the general

area of control of large space structures. However, there is a need at

this time to investigate the dynamics and control problems specifically

related to the Satellite Power System to assess performance of selected

control concepts, and to identify and initiate development of advanced

control technology that could enhance feasibility and performance of the

MPTS stands for Microwave Power Transmission System.

n p

1

SPS syctotm. Two of the areas that have been under investigation are

the dynamics and control of the solar collector with the MPTS antennae

treated as point masses and the dynamics and cvatrol of the MPTS antenna

with the solar collector treated a$ a dynami4 disturbance source. This

report covers the most basic problems of the latter, that is, the dynamic

properties and the performance characteristics of ;:he MPTS antenna.

It is a known fact that the losses accrued at the later

stages of a process are more costly than those accrued at the early stages.

This is also true in the context of the power collection-conversion-

transmission process of the SPS system. Since the antenna power trans-

mission constitutes the last part of the efficiency chain for the in-orbit

operation, a great deal; of emphasis has been made directly to the pointing

accuracy. however, due to the high weight penalty at synchronous altitude

and the huge array size (1000-meter diameter), the structure cannot be

made arbitrarily rigid, and the structure stiffness and hence the surface

deformation of the anterzu,l plays an important role in the determination

of the performance of the antenna. It is this latter subject area which

this study is focussed on.

1.2 PERFORMANCE REQUIREMENT GUIDELINE

In References 2 and 3, the requirement of mechanical pointing

and alignment accuracy of 2 to 3 arc minutes has ''jeen considered. In

this report, the performance requirement of 3 are minutes for the surfaceflatness is used as a base for discussion.

1.3 OBJECTIVES AND SCOPE

The combination of flexibility, huge dimensions, high distur-

bances, and the nature of the operation make the MPTS/SPS antenna uniquely

different from any antenna or spacecraft that the aerospace community has fi

ever designed. The objectives of this part of the study have been toi,

establish the basic dynamic properties and performance characteristics

of the antenna so that this knowledge can be utilized to form a new base,

requirement, and constraints to aid the control system design. Specifically,

the following subject areas have been addressed based on the most up-to-date

MPTS data,

2

y{

1, The vibrational properties such as the mode shapes

and natural frequencies of the antenna and their

effect to the SPS structure properties as a whole.

''xi The surface deformation of the antenna structure and its

effect on the scan losses,

3. The effects of disturbances including thermal distortions

of the solar collector to the flatness of the antennasurfaces

4. Application of the linear analysis technique to extend

the results obtained through simulation,x

The approach employed here has been one of the time domain

analysis techniques, i.e., a combined modeling, analysis, and computer

simulation approach.

In Section 2, the necessary mathematical formulations of the

model and the performance related terms and quantities are presented. The

-tzuctursl Flexibility properties, the eNtirm,Mtes of the disturbances, the

thermal effects, the dynamic interactions, and the main results of the

simulation and their linear extensions are presented in Section 3.

1.4 STUDY RESULTS AND CONCLUSIONS

The main contribution of this part of the study is that it has

established the performance characteristics of the antenna, especially the

local warping and its effects on the antenna scan losses. The performance

characteristics are plotted as time histories of a number of quantities and

they have also been summarized in tabulated forms in Section 3. In

addition, these results have been extended in the parameter space as shown

• in Figs. 23, 24, and 25 (these figures are reproduced here for quick

reference).

Throughout this study, a great deal of insight and experiences

have been gained such as how certain types of vibration modes react to

given types of signals (forcing functions) and their position of execution;

how modal dominance varies with the frequencies and the properties of

the forcing functions; the importance of signal shaping and timing; the

distribution of modal energy and its time dependence, etc. All

of these are invaluable information for the control system design.

17

3

4

Computer simulation programs have been developed which have

proved to be an effective and necessary tool for this study and it will

be an essential part of the facility for our future work. These programsare listed in Appendices A and b for reference purposes.

The major conclusions of this study are as follows;

1. The structural damping is essential for the maintenanceof a flat radiation surface. This is especially important

for dissipating the otherwise prolonged presence of highsurface transient caused by disturbances. The decay rate

of the surface deformation in general agrees with that of

the amplitude of the dominant mode. A structure damping

ratio of .005 has been used; a higher value will be more

desirable.

2. The most significant sources of disturbance that affectthe antenna performance perhaps come from the dynamic and

control interactions between the antenna and the collector.

The effects of these disturbances may be reduced significantly

by

a. 01,4coupling the antenna from the collector motions,

especially the translational motions;

b. actively controlling the collector bending motions

at the interface boundary. Large amplitude

oscillations at the collector tips must be Actively

suppressed.

Figures 23 and 24 show the performance characteristics of

the antenna surface as a function of the amplitude of

oscillation (cosine function) for collector bending and

torsional motions, respectively. A suddenly applied

bending oscillation of slightly over 1/4000 of the

length of a 20 km collector will cause a maximum RMS

local slope angle of 0.05° or 3 arc minutes.

I

4

3. External disturbances such as the gravity gradient, solarpressure, etc., will not have a direct impact on the surface

flatness with the possible exception of the thermal distor-

tion of the collector structure. High temperature gradient

with short thermal lag time after shadowing, can set thecollector to sudden beading oscillation. Unless low

coefficient of thermal expansion (CTE) graphite composite

material is used, this could be a source of serious problems.

fiFigure 25 shows the estimates of the collector bending

amplitude as a function of the CTE for a number of tempera-

ture gradient values. By using this estimate with the

performance data of Fig. 23, a suitable CTE value may bedetermined,

4. Signal shaping and timing are of critical importance to theperformance of the MPTS/SPS system. The shape of an

actuator force Fhould be designed so that the least amount

of energy may be absorbed by the flexible modes. The cut

off time of a thrusting force determines the amount of

energy left in the flexible modes. To reduce the level of

surface vibration, the modal state of the dominant mode

should be monitored so that the cutoff of a major thruster

should be timed such that the modal energy of this mode is

at its minimum,

5. The MPTS/SPS has the following structural properties:

a. The natural frequencies of the antenna and the

SPS are 2 to 3 orders of magnitude greater than

the orbital frequency. That means that these

modes and orbit will not couple.

b. The frequency of the lowest flexible mode of the

antenna is about one order of magnitude greater

than that of the collector. However, the first

3 flexible modal frequencies of t1jie MPTS overlapwith the 8th of the collector, at3d the excitation

of the latter should be avoided.

5

f

6

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4

3

2

a. Due to its location, the antenna mass will have moreinfluence on the SPS normal frequenrl"o-s than the samemass distributed over the solar collector. The antennamass is about 30% of the total SIPS mass, its existencehas decreased the fundamental frequency by a factor of0.4 and its effect on the higher frequencies is less(5). The stiffness of the coupling structure alsoaffects the normal frequencies. Tile significance ofthese effects to design alterations are yet to bedetermined.

d. Tito most significant factors that affect the modalproperties are the geometrical parameters of thestructure.

AMPLITUDE OF ACCELERATION FORCE, N

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Figure 23. IIPTS Surface Deformation vs. Amplitude ofCollector Bending Oscillation

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SECTION 2

DYNAMIC MODEL AND PERFORMANCE RELATED EQUATIONS

2. 1 T11% DYNAMIC EQUATIONS AND ENERGY OF CONCENTRATED MASSES

Dynamic Equations

in Reference 1, fini,u t^l.c ment models of Lhe MI'L'S antenna

have been discussed. In this sectlot+ only the key elements of the model

that have been implemented in the simulation programs are discussed.

The model consists of 20 decoupled second order differential

equations representing the modal amplitudes of the 6 rigid body modes

and the 14 lower order flexible monies,

1!tq + Kq - 0Tf (1)

where q is the 20-modal amplitude vector; f, the 498-force vector; M and

K are the 20x20 generalized mass and generalized stiffness matrices (boat

diagonal), respectively; and (P is the matrix of eigenvectors of dimension

498x20 « For convenience, (1) is written in the following form,

1.

q `+' Aq u ^Tf (2)

where

A - R 1R wk2 (3)

where wk is the angular frequency of the kth mode for k-7,...,20 and

Wk : 0 for k-1, ,6; and

S(M-1)T . 0 M-1 (M is diagonal) (4)

In this study, structural damping is also considered. Since there is

no available information, a damping term, with 4 = . 005, has been added

to the kth flexible mode as follows, in the expanded :form,

q $kT Mk-1 f kffi1,...,6 (5a)

8

q + 2Cwkq + wk2q in 'l Mk7l

f, k-7 , ... v 20

(5b)

where 0k

is the kth column of 0 and Mk-1 is the kth diagonal element of

Mk-1 Since these equations are linear time invariant, analytical solutions

are readily known. Compact solutions were therefore used in the simulation

programs instead of the more time consuming methods of numerical integrations.

In the following, let T be the interval of each computation step, and

a wk /1-^ 2 T, C^ . cos y , S^ • sin + , and q (n) ^ q (nT) , then in eachcomputation step, the Following is eva luated,

a:

For k - 1,.. . ,6

q (n+l) 1 T q(n) ] + T'/2 ^k Mk f (n) (6a)

q(n+l) 0 1 j(n) T

For k 7,...,20

q (n+l) C,. +-^4 , S 1 S q (n)'

= e4wnT 3177 ^ wk' 1-4 ` ^

q(n+l) - wk S hy ^- S^+ C q(n)31-7

1 - e-4WkT (C + 4

S )

+ wx2 WOk rlk 1 f(n) (6b)

k-^°k-^- S

where f(n) is the force vector evaluated at t - nT and n - 0,1,...,tf/T;

t f is the final time.

With the availability of q k , the displacement of node i,

i = 1,..,,166, due to ela stic deformation alone, at t = nT, can be

evaluated as the linear combination of q k, i.e.,

20

ui(n)

X ski qk(n)k*720

vi(n) kI7

^k(166+i) qk(n)

202C

wi(n)k7

Ok(332+i) qk(n)

and the displacement due to the rigid body modes,

6

ui(n)kl ski qk(n)s

°i 0) - k l ^k(166+i) qk(n)

wi (n) 1 0k 332+1 qk(n)kMl ( )i

The actual position of node i, at t a nT, is

xi (n)- Xi + ui (n) + ui(n)

Yi (n) - Yi + v

i (n) + ui(tt)

zi(n) - z + Wi (n) + wi(n)

where Xi , Yi , Zi are the coordinates of node i when the system is at rest.

Equation (9) was used to generate the antenna surface and the local slopes

with the rigid body motion suppressed, i.e.,

xi (n) - Xi + ui(n) (10a)

Yi (n) Yi + vi (n) (10b)

;. i (n) = Z + wi (n) (10c)

where (xi (n), yi (n), zi (n)) may be interpreted as the coordinates of node i

at t - nT projected onto the body frame, whereas (xi (n), yi (n), zi (n)) is

that expressed in terms of an inertia frame.

Energy of Concentrated Masses

The body energy of concentrated masses is of great interest to

this analysis since it is a good measure of modal activities of the antenna

10

kt

at

..

(7a)

(7b)

(7G)

(8a).r

(8b)

(8c)

(9n)

(9b)

(9c)

under disturbance. Let E be the sum of kinetic and potential energy of

the antenna. Recall that the antenna is modelled as a collection of

concentrated masses which are connected and having finite stiffness. E

may be expressed

E _ 2 xTmx + 2 xTkx (11)

Since x - Oq and M = OT m 0, K = @T k 0, (11) becomes,

E QT_q +2 gi1 Kq (12)

Recall, that M and K are diagonal. E may be written in terms of its modal

components,

Ek - 2 Mk qk2 K 1,...,6 (13a)

1 ' 2 2 (13b)Ek 2 Y' k + wk qk-), K 7,...,20

Let ER and E be the rigid body energy and the flexible body energy,

respectively, then

6ER - 2 Mk qk2 (14a)

k 20

EF = 2 Z rik (gk2 + wk Qk2 ) (14b)

k=7and

E - ER + E (14c)

Since q and qk are computed in each step, the energy Ek,

ER , EF , and E can be readily evaluated. It will be illustrated later

that the modal energy is indeed a good measure for determining the relative

dominance of the modal activity. Further, it may be used as a measure for

signal shaping and thrust on -off timing.

2.2 .LOCAL SLOPES OF THE SECONDARY STRUCTURE

The basic architecture of the MPTS antenna consists of two

parts, the primary structure and the secondary structure (Fig. 1). The

11

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primary structure is a tetrahedral planar truss structure which serves

as the base support to the system and provides the required structural

stiffness. The secondary structure is a "finer" structure. which is mounted

on top of the primary and it provides a base for the installation of

subarrays of the antenna. Due to the huge size (1000 in dia. x 130 in depth- nominal) and the weight penalty of a satellite to be built and trans-

ported to a synchronous orbit, the stiffness of such an antenna cannot be

made arbitrarily high. A series of questions may be asked, such as,

(1) how stiff the structure will have to be in order to meat the operational

requirement?; (2) is the configuration under study stiff enough?; (3) is

there any tradeoff between structural stiffness and active shape control?

Before these questions can be addressed, the relationship between the

structural deformation and the w.^tenna transmission loss must be established.

This is the subject of the next subsection. In this subsection, the local

slopes of the antenna radiation surface is derived.

For structures of finite stiffness, surface deformation will

occur if disturbances or forces are applied to the structure. Accompanied

with the local displacement is the change of the direction of the normal

to the local surface, or the local slope. Since the antenna is retro-

directive phase conjugate, the transmission efficiency is not sensitive

to the variation of path length or the surface dispositions but it is

very sensitive to the local slope changes. Therefore, it is important

to evaluate the local slope variations.

Since the finite element model being used does not provide the

details at the subarray level, one can only compute the slopes of the

secondary structure. In fact, the secondary structure is not modelled

either; its slopes are computed by assuming that the local surfaces are

Flat and each one a part of one of the 61 plane surfaces. Referring to

Fig. 1, the secondary structure consists of 61 hexagonal substructures

referred to here as the hexagonal planes. Each hexagon plane is supported

at the three symmetrical vertices by the supporting pins on the primary

structure such that there is no direct contact between the primary and

the secondary structures and that there is no contact between the secondary

14

r

substructures themselves. This is done to reduce the interactionsbetween structural components. It may also be importance to point

out that the grid points assigned in the model coincide with the

supporting pins (actually each grid point represents ti ►e threeclosest pins and in positioned at the geometrical center of the three

pins (refer to figs. 2 and 3)) .

With this background in mind, the local slope angles can be

readily derived. 'there are at least two ways for computing the slope

angles, (1) using the equation of a plane, and (2) using vector product.

Let (xi t yl , z1)0 (x20 Y20 zg). and ( 3

0 y" z 3) be the

coordinates of the three grid points associated with a typical hexagon plane

i (tile index i and the grid point index j were dropped). In the ,following

the bar in ":" that signifies the grid point position with the rigid body

motion suppressed is also dropped.

Method 1 - Plane Equation Approach:

Let a, b, and c be the constant coefficients, and the equation of

a typical plane may be written as,

ax+by+cz . l

(15)

The coefficients may be determined by substituting the three points into

(15) and solving the three linear simultaneous equations, or

ax yl

z1 -1 1

b x2 y2z2l (16)

c x v3z5 1

To avoid numerical iterations, compact expressions for the matrix

inversion may be used. Since the coordinates of the three points are com-

puted at each step, the values of a, b, and c can be evaluated accordingly

and the slope angle error, s, is readily obtained,

Q - cos-1 2 10 —,f (17)a +b'+c

15

idl

60;

45

2025

45

SECONDiTRUSS

a 1

SURFACESTRUT Y NODE

POINT

(a) 75 Grid Paints on Array Side of Primary Structure (only 13 withIA shown),

SUM PORTING PINS

rKIMAK T i KW,J

(b) Secondary Structure Supporting pins.

Figure 2. Grid Points of the Finite Element Model and theSecondary Structure Supporting Pins

I

Y

Figure 3. The Grid Points and the 61 Hexagonal Planes

X

Figure 4. Plane Intercepts

17

Note that the matrix inverse in (16) does exist, since, in our applicationthe three points are always non-colinear. However, this meVt°ad could leadto computational difficulties. Referring to Fig. k, 1/a, 1/b, and 1/0

are the three intercepts of the plane on the X-, Y-, and 2 -axis,respectively, hence, as the plane becoming parallel to the XY-planea -r Q, b -). Q and c Since 4 is a very small angle (and time varying),double precision computation will be required for the evaluation of (17).

Method 2 - Vector p roduct Approach;-

This appraoch is simpler than the first method and it requiresfewer computation steps, Let Al and A2 be defined as follows,

x2 - x

Al " y2 ` y1 (18a)

z2 z

x3 - x1

A 2 " YS - yl (18b)

z3 z1

Let their vector product be (A B C) T , then

A

B Al x A2 - Al A2(19)

C

where

Q-X3

X2

x X3 Q ^x1

.,,x2 x1 0

the slope angle is

8 cos -1 ICI . (20)3A` +B +C

18

3

1 ,4t

m:r

For the same reason discussed in Method 1, double precision computation

will also be required in evaluating (20). Method 2 was used in our

computation.

The slope angle, 0 1 (n), is a function, and its value varies

with time (n) and space (i). The individual. values of the slope angle

will not mean much for the antenna as a whole. Other performance measure-

ments must be defined. Three measurements are used Mere.

The Maximum Slope Error

0M(n) = Max 01(n) (21)

is{1, .... 61}

0M(n) is the greatest slope angle of the 61 hexagonal sub-

structures, at time t = nT; therefore, it represents the worst case.

The RMS Spatial Average

6 RMS (n) - 61(01(n))2 (22)

i=1

0 RMS (n) fluctuates with time which is a good measurement of the antenna

at a typical time t = nT.

The RMS Time Average

RMS - Fn (a (m)) = ^16^*(6 (m)>2

(23)m-1 61n m=1 i=1 i

ORMS(n) is a smoother measurement especially when n becomes large. This

measurement is useful when one asks how is the antenna doing so far. 0 RMS (n)

becomes less and less sensitive to the current sample as n increases,

which is a common defect of the averages of this kind. This problem may

be solved by applying a window to the samples, i.e., one only uses the N

most recent samples in the average. Another way to increase the sensiti-

vity is to assign a heavier weight to the most current sample, for instance,

assign 10% weight to the most current sample and 90% to the rest as a

whole when n > 10. This latter approach does not require additional

storage space.

%I

19

2.3 THE MICROWAVE TRANSMISSXON SYSTEM SCAN LOSSES

Since the design of the HPTS antenna is still in the evolving

stage, the data and assumptions made in this section are far from final.

Nevertheless, the results of this and the subsequent sections will still

be valuable Information for the purpose of developing performance require-

ments.

The MPTS antenna is a phase array antenna which consists of

a large number (say, on the order of 10,000) of array elements. Array

elements are installed on the 61 secondary substructures. Each element

has an aperture area of about 108 m2 , or for a rectangular aperture the

aperture size is about 10.39 m.

The array is active retrodirective so that it is made insensi-

tive to the path length variations but it is quite sensitive to angular

deviations from the line-of-sight. The latter requires accurate pointing.

Pointing will be achieved by two control systems, the mechanical pointing

control which is required to stay within a few arc minutes of the target-

the rectanna, and the fine pointing control-electronic beam steering which

is required to steer the beam to within several are seconds.

Let D be the aperture, 0 be the slope angle (or angular pointing

error), and X be the wavelength of the microwave power. Since the frequency

is 2.45 Gllz, the corresponding X is .1224 m. For uniformly illuminated

antenna with rectangular aperture, the radiation pattern is

sin {I sin0) sin { 0iTD sing

m

D 9

the approximation is valid for small 0. Equation (24) prescribes the

patterns for both the array and the elements, for the array, one substitutes

Da m 1000 for D and for the elements, use De = 10.39 for D. 'From this one

can see immediately that the array pattern is much narrower than that of

the element. More precisely, the half-power-width (the beam width at the

half power points) of the array is about 100 times narrower than that of

(24)

20

a

P

the element. At the half-power-point nDO/X - 1.3916 rad and the corres-

pending 6a and O e are .0031* and .299% respectively. Figure 5 shows the

antenna and element patterns.

In the following it is assumed that the electronic beati►

steering system points the beam at the rectenna perfectly and the

discussion is concentrated on the effects due to surface deformation of

the structure, i.e., the effects of scan angle variations on

the antenna

efficiency.

t.Scan Loss

The amplitude of the array pattern varies with the scan angle.

The field strength decreases rapidly as the scan angle increasingly

deviates from the normal of the array surface. The governing relationship

is that the array beam amplitude is prescribed by an envelope which is

determined by the contributing element patterns. In

the case that the scan

angles are the same for all the subarray elements, the envelope becomes

the element pattern itself. This is illustrated in Fig. 6. The loss of

power due to scan is called the scan loss.

Let 11T be the total power, Pi the maximum power of element i,

and sin x e /x e

the corresponding element pattern, where the argument

Xe a if De Q 1/X, then

s in x 1 2P T x

(25)

a. Rigid array surface

in this case all x x or 0 , W 0

1,

sinx 2 sinx 2P,

T x x TM

where P TH

is the maximum power of the array. The scan

loss 1'L and the percent loss are,

PL PTM [l

Slinx 2l 1x

sinx 2P z '—") ) x 100L x

(26)

21

—01 6-11 4-^ 0.000245 rod = 0.0140DoDo

;:;

R^ = 0.0236 rod t:: 1.359Do De

Figure 5. Array Pattern and Element Pattern for Rectangular ApertureArray with Uniform Illumination

I —

, EFF)

no D CrAM ANGLE

f f f

rZl

0.0070 0.3350 0.675°

Figure 6. Beam Amplitude vs Scan Angle

,WER BANDWIDTH

'ATTERN ; I

22

r

The percent scan loss is plotted in Fig. 7 which

illustrates how critical the scan angle is to the power

output.

b. Uniform illumination

For uniformly illuminated array, P i P,

rr sinx P N sinxP= P L ( i) 2 = ( Tm) G (

i) 2

T i xiN i=1 xi

where N is the total number of elements and1 1 sinxi 2

P _ Cl - ( ) ] x 100 (27)L N k

=1 xi

c. The RMS approximation

Assume that the root-mean-square value of the scan angle,

8Rr,S, is known, where

1 N 2

AIMS Nitsi

The corresponding xRMS can be easily evaluated:

xRMS = 3 N ( e î)2 = (^ e)

'RMS (28)i=1

then nD 2,N sin (-

X -E

SPMS)

PTRMS i t Pi 1tDe

A 13RMS

trD 2sin (

A^ SRNiS)

fD PTM (29)eX S RMS

and the percent scan loss is,

^rD 2

sin ( ê SRPiS)PLRMS%

1 - ,Dx 100 (30)

ea eRMs

Note that in general PTRMS f P T and hence PLRMS 0 PL'However, our simulation shows that for many test cases

23

M%

80

70 /. 1

60

50

ZpLnLn

40

Zl<ULn

30

20

10

0I^/ 1 1 1 J

0 0.1 0.2 0.3 0.4

SCAN ANGLE, DEGREES

Figure 7. Scan Loss of a Rigid Array

,,

r s

24

a.

t- i

4

they are quite close. Therefore, PLRMS%

may be used

instead of Ph%. The former requires much fewer computation

steps. Note that the 0RMS

here is the RMS spatial average.

d. Gaussian illumination

in order to reduce the side lobs of the Array pattern,

the elements may be illuminated non-uniformly. One way

to accomplish this is to tailor the illumination of

the element according to a Gaussian Curve, In this case,

P = Po e-(R/Ro)2 a (31)

The parameter a may be determined by the taper at the edge

of the array. For a 10% taper, i.e., F/P o M 0.1 when

R - Ro , a - 2.3. A 10-step taper that approximates a

Gaussian curve was proposed in an earlier study [2].

From (25), PT becomes,

PP e-(Ri/Ro)2a (sinxi)2 (32)T o

ii

sinxi

)

2 -(Ri 2a

eo i xi

where R is the diatance from the center of the array to

the element, and R the radius of the array aperture.

of the last three cases, only the first two were implemented

(refer to Appendix A).

Y

25

SECTION 3

SURFACE DEFORMATION AND ITS EFFECTS ON THE ANTENNA EFFICIENCY

DUE TO DISTURBANCES AND DYNAMIC INTERACTIONS

This section addresses the problem of surface deformation

and its effect on the microwave power transmission efficiency in quanti-

tative terms. Specifically, the question of how significantly the disturbances

will impact on the antenna l s flatness and how much disturbance the antenna j

can stand without incurring excessive power loss will be answered. The Ii,

results were obtained through extensive computer simulation.

Before the main results are discussed, the modal characteristics

of the antenna and how these affect the overall system modes is briefly

described and then the magnitude of disturbance and dynamic interaction

forces are estimated which form the basis for the magnitude of simulation

input.

3.1 THE MODAL CHARACTERISTICS OF THE ANTENNA AND THEIR EFFECTS ONTHE OVERALL STRUCTURE OF SPS

The fundamental properties of a large space structure such as

the MP'TS antenna that have significant effects on the control system design

are the structural vibrational properties which are governed by the funda-

mental frequencies and the mode shapes. The former affects the controller

bandwidth design and the latter affects the sensor/actuator location

selections.

The first ?a natural frequencies of the structure are summarized

in Table 1. Of the 20 modes, the first 6 are the rigid body or zero

frequency modes and the rest. are flexible modes. Figure 8 shows the

mode shapes of the first 14 flexible modes. The first two modes in Fig. 8

exhibit astigmatism bending, the third one is a defocus mode, and the

fourth mode (mode 10) exhibits trefoil bending characteristics. The rest

of the flexible modes have either the similar modal characteristics of

the first four but with higher frequency patterns or a combination of

them.

26

T,

1

I

Table 1. The First 20 Model Frequencies of the Antenna Structure(Data obtained from Ref. 4).

1 0 Rigid Body Made

2 0 to

3 04 0 it

5 0 10

6 0 it

7 .0189 Flexible Modes

8 .0190

9 .0196 it

10 .0343

7,1 .0345

12 .0348 of

13 .0430

14 .0457 it

15 .0463 to

16 .0471 to

17 .0499

18 .0505

19 .0515 ' ►20 .0517

I'

r

28

a

' z

}

t-

r

I

k

rli

J_ _

W

u 0OCG

b

`=9

x,.: W

KOK

w

a^

a,

0v

W

Or1

1J

HMiW

Cl

uwo

n a^

WQ cn

aW

x W

J

W

0

O

00JW0

OC6

inJW

O

O

Jw0O

MJW0O

JW0

0NJW02

. I

^JOO

O_JW0

E

The mode shapes of the first four flexible modes are in

general agreement with that discussed in Ref. 3 where only the first

four flexible modes were illustrated, However, the modal frequencies

are about k to 7 times greater in Ref. 3 than those listed in Table 1.

It is also noted that the nominal mass used for the antenna in our model

is 15 x 106 kg whereas it was 8.58 x 10 6 kg in Ref, 3 9 a factor of 1,75.

Since the generalized stiffness is proportional to the mass times

frequency squared (mw 2), the stiffness is About 11 times greater for

modes 7, 8, and 10, and greater for mode 9 in Ref. 3 than these in our

model.

Figure 9 shows a comparison of the natural frequencies of

the WTS antenna, solar collector, and the first five frequencies of the

flexible modes of the SPS. The following conclusions may be drawn.

1. The fundamental frequencies of the SPS and the antenna

are 2 to 3 orders of magnitude greater than the orbital

.frequency. This means that the couplings between the SPS

flexible modes and the orbit will not be significant.

2. The fundamental frequency of the antenna is about one

order of magnitude greater than that of the collector.

However, the frequencies of the first three flexible

modes overlap with that of the 8th mode of the collector

which indicates possible couplings between these modes.

Due to the fact that higher frequency modes are less

likely to be excited, the likelihood of their exciting

the lower modes of the antenna are also reduced.

Nevertheless, caution must be exercised in the controller/

actuator design so that mode 8 of the collector will not

acquire excessive energy at any time.

3. The antenna does not decrease the SPS :frequencies signifi-

cantly nor does the coupling stiffness [5]. This is

because of the superior size and mass of the collector

over that of the antenna. The implications of this

29

,x }r

observation is that the design alterations of the

antenna and the coupling will not significantly impact

on the vibrational properties of the overall system.

4. The most significant factors Lhat affect the modal proper-

AA0 ties are they geometrical properties of the structure for

both the antenna and the collector, i.e., the size and

the relative size (e.g., aspect ratio) of the dimensions

of the structure,

The 3-D plots of the mode shapes was done by the program listed

in Appendix R and the required dynamic responses were computed by the

program listed in Appendix A.

3.2 ESTIMATES OF DISTURBANCES AND DYNAMIC INTERACTIONS

Disturbances may be categorized as external and onboard.

External disturbances include gravity gradient torques, solar pressure,

magnetic torques (due to the interaction between the Earth's magnetic

field at the synchronous orbit and the onboard current loops), etc.The onboard disturbances include, for instance, the microwave recoil,

dynamic unbalance torque, current interaction torque, thruster firing,

etc; Qualified as both external and onboard is the thermal bending distur-

bance due to temperature gradient of the structure. Excluded from either

category is the dynamic interaction between the array and the collector

such as the forced motion of the antenna due to collector 'bending.

The effects of some of the disturbances can be reduced by

careful design. For instance, the current interaction torque may be

reduced by careful arrangement of the conductors so that the conductor

pairs are coplanar or by using coaxial. Cables. Thruster impingement may

be reduced by using gimballed pairs and by careful selection of their

locations, etc. Solar pressure and microwave recoil forces (and torques)

are rather small to cause any flatness problems of the antenna. They

may create station keeping problems more than anything else. Therefore,

it is sufficient only to look into the effects of gravity gradient torques

and the forces due to collector bending.

30

1

Gravity Gradient Torques

The gravity gradient torque about the x-axis (refer to Fig. 10)

Tx. ni.1 (^^^ (Iz '.'. Iy) sill 2 (%: 0x)

.. _ ^ wo2 (1z - Iy) sin 2 0x (33)

where x-, Y-, and z-axes are the body axes of the antenna, x and y are

the inplane Axes and z-axis is normal to the array surface. I and Iz are

moments of inertia and their values are,

Iy M Ix - 7.24 x 101 '1 Kg-m2

Iz w 1.45 x 10 12 Kg-m2

and the orbital rare w is 7.272 x 10 -5 rad(sec.

Assuming that the orbit is inc lined at 7.5°, the satellite

will oscillate of + 7.5° about the equatorial plane once a day (refer to

Fig. 10). For a rectenna located near the equator, the attitude of the

array will have to vary ± 1.34- about the local vertical once a day.

For a rectenna located at 45°N the angular deviation will vary between

6.04* and 7.53° from the local vertical. For an equatorial orbit, the

angular deviation will be a fixed bias of 6.83 0 for the 45°N rectenna.

Of all these cases, the maximum deviation angle from the local vertical

is 7.53°, the maximum bias angle is 6.83°, and the maximum cyclic angular

amplitude is 1.34°, and the corresponding gravity gradient torques are,

respectively, from (33), -1496 N-m, -1360 N-m, and -269 N-m. These torques

are instabilitizing disturbances, i.e., @x = 0 is not a stable equilibrium.

In all of these cases, control effort will be required to offset

gravity gradient effects in addition to tracking (attitude

correction to compensate latitudinal motion due to nonzero inclination

and to compensate longitudinal motion due to nonzer6 eccentricity).

The control problems will be discussed in Volume II.

The problem of array flatness is the main concern here. It will be clear

is,

r

31

M PTSANTENNA

COLLECTORSTRUCTURE

COLLECTORWITIEND-MASSES

)6 Hz

Figure 9, Modal Frequencies of the MPTS and SPS

N RECTANNA t Y

_ c ... z Xg 6.04 7. :..: —

4 6380 Km45° 7.50°

,^ Y

7.50

EQUATOR 42180 Km^ :.. # 9 ^ 715

tea. _ ^ f Y

1.340_ z

xs

Figure 10. Gravity Gradient at Three orbit Positions ofthe MPTS/SPS Oscillating with a 24-hourPeriod.

* I

later that the warping of the array surface due to gravity gradient is

relatively small and insignificant due to the relatively low disturbance

level and the low frequency nature of this torque.

Collector Bending Forces

Collector bending motion may be caused by many reasons, for

instance, thermal effect, thruster firing for station keeping, maneuvering,

attitude acquisition and control, etc. Unless the translational and

angular motions of the collector at the interface boundaries are controlled

or minimized, they may be the most significant causes of the antenna

warping and power loss.

a. Collector bending due to thermal effect

In order to estimate the magnitude of the thermal bending

distortion, a collector configuration must be assumed.

To be consistent with the work of Johnson Spare Center (5)

and for the purpose of comparison, the following analysis

employs a full configuration identified as 20D4 in Ref. 5.

Let R and d be the length and depth of the collector

structure, respectively, and their values are, A = 20,000 m,

d = 400 m. Assume that the temperature is uniform on each

section parallel to the collector surface and the tempera-

ture gradient is uniform along the depth. Let AT be the

temperatufaa difference between the front (solar blanket side)

and the back surfaces of the structure. Let a be the

coefficient of thermal expansion and ke be the displacement

of the antenna interface from the boundary of the collector

structure. Under these simplified conditions, the bending

displacement, H, is,referring to Fig. 11,

H = aAT (1 + a2T) ^l - cos(R2dT)

+ V sin(RaOT)2d

(34)

The displacement with respect to the center of mass, H c , is

He = .444 H

(35)

P.

33

'M l

d

o

b

as

N0u

î

H

P4

Ln

(hOtiZ

"

o^

,.a

q

34

r

V

For a temperature difference AT - 100% the corresponding

values of 11 find tic for throe: values of a ar ► shown, in

Table 2.

Tile coofficiaa^ of 10-5 Pl/m/'C is in the lower range ofmetal. For instanne, pure aluminum has a thermal expansion

coefficient of 2.36 x 10-6 /'b C at the temperature range of293 X to 393 K and iilightly less for aluminum wrought alloys.

The values of 10-6 and 10-7 are in the range of graphite

composite material.

The thermal disturbance oil the solar collector is the worst

at the vernal, and the autumnal equinox where the ST'S

will stay in the full shadow the longest period of time.

The thermal transient occurs when the collector moves into

the shadow .and again when it moves into the sun light from

the shadow. Since the trAnsit time during the penumbra is

about 1.63 seconds, which is about 1/570 of the period of

the first bending mode of the SPS structure, the effect due

to the penumbra, is ignored. For a similar argument, the

thermal Ing time 'is also ignored since the 'Lag time is

rather small due to the sparsely distributed structural mass.

As a result of these idealized asswiptions, one may construct

this scenario. The tips of the structure tire bent away from

the sun while tile structure is hearted by the full sun, and

the equilibrium condition is renehed. Suddenly the collector

enters the shadow and the temperature gradient is

reduced to

an insignificant value, and the distorted structure isreleased from its retension force and starts to bounceback. This causes

all oscillatory motion at they tips

about the new equilibrium state. This motion applies a

sinusoidal cosine translational acceleration or forces atthe antenna interface.

35

A

'fable 2. Beading Di.splece"nt

..r 0

a, m /m /^C 11, n► 1[c, m

10`5 150.0 66.66

10-6 15.0 6.67

10-7

115 .67

Tabl4 3. Thermal Disturbance Estimates

CL , n► /m/°CZ III , n►

A III ,u► /s z Amp 8 rIII, N

10-5 66.66 3.037 x;1,0-3 .31 x 10-3 45558

10-6 6.67 3.039 x 10-4

.31 x 10 -4 4558.5

10-7 0.67 3.039 x 10-5 .31 x 105 455.9

A

:a36

The above scenario is drawn to help in the determination

of the proper forcing function. The real interest here

is not in the period when the SPS is in the occultation,

but is to the period after the occultation. With the

above discussion, one can construct a similar but reversed

scenario; that is, the structure starts a bending motion

after it enters full sunlight. The only difference is

that this time the equilibrium surface is a bent surface

whereas in the shadow it is a plane.

Let Z be the displacement of the antenna interface, then

Z = -Zm cos wst (36)

and

Z ws2

Z cos ws t (37)

=A cos wtM s

and the equivalent force applied at the antenna :interface

will be,

F = ma w s 2 Z cos wst (38)

where Z = Hc , w = 21r, and fs = .00675 rad/sec and is the

first bending frequency of the SPS. m a 15 x 106 kg is

the nominal mass of the antenna. Table 3 shows the values

of Z , Am, and Fm for the three values of a.

The effects of the thermal disturbance on the antenna's

surface deformation are examined in the next subsection.

b. Collector bending due to other disturbances

The collector structure bending caused by disturbances

other than thermal effect may be significant. Since an

accurate estimate requires the details of specific designs,

therefore, instead of being specific about the disturbances,

a more generic approach is taken here. Since it is not

inconceivable to anticipate a Z of 1/2000 of the length

of the collector structure, forces corresponding to a

37

. .r

lk I

range about this 2m were considered in the simulation.

Both sinusoidal and step functions were applied.

Torsional Notion of the Collector at the interface

y

The first torsional mode (see Fig. 12) of the SPS is the third

flexible mode which has a natural frequency of .00359 Hz or .0226 rad/sec (5).

For a V torsional oscillation at the collector's tip, the correspondingtorque applied at the antenna interface will be,

}

r = I w2 S. 6.4526 x 106i N-mm m

The corresponding force exerted at the antenna interface will be

Fat - 6452.8N.. Sinusoidal and step forcing functions were used in the

simulation. Note that a V torsional oscillation at the tip of the

collector corresponds to a translational oscillation of 43.6 in at the

extreme boundary (point a in Fig. 12) and 8.7 m at the antenna interface,

which is intolerable during the normal operation of the antenna although

it may occur during maneuvering operation.

3.3 SIMULATION, RESULTS, AND DISCUSSION

3.3.1 Types of Forcing Functions

The performance of the array structure was tested (via computer

simulation) for a number of signal types that approximate the disturbances

or maneuvers it may encounter during operation.

The forces and torques applied are step functions, rectangular

pulses, and sinusoidal functions. The first two kinds are for maneuvers

and the sinusoidal forces are for simulating dynamic interactions induced

by collector boundary oscillations. In this later case, both sine and

cosine functions were applied, the former represents a geadually applied

cyclic acceleration whereas the latter, a cyclic shock load.

In all the cases, the forces were applied at six positions on

the primary structure as shown in Fig. 13. At each of the grid points

38

00 m

AQ

b

i

yt kW

T

Figure 12. The First Torsional Mode of the Collector

39

604;

6025704,

>.0452025

1045

Fd/Fd

Fd/,

:d/2Fd

;d/2

Fd/

Fd

Fd/.

-Fd/2-Fd

.Fd/2

(b) MOMENT

(c) GRID POINTS

Figure 13. Positions of the Applied Forces

(a) FORCE

6025 and 2025 the force Fd was applied and at each of 6045, 7045, 2045and 3045 the force Fd /2 was applied. Although the 6-point distributedcase is more realistically representative of the antenna and collector

I interface, the results show no significant difference from 2 points

(6025 and 2025). Torques were simulated as couples. All the forcesconsidered are normal to the antenna sur ac4; therefore, the results

represent the worst case. Note that the antenna/collector interaction

will have the greatest effect on the antenna surface when the normal

of the two surfaces are in parallel.

3.3.2 Simulation Time Considerations

In most of the cases the simulation time of 1000 seconds

was used. This time covers at least one complete cycle of the lowest

bending mode of the system (the longest modal period is 930.8 sec.). For

the torsional acceleration, 600 seconds were simulated which covers

more than two complete cycles of the excitation torque (the first torsional

period is 278 sec.)

The computation step sizes of 2.5 seconds and 5 seconds wereused which represent about 1/20 and 1/10 of the dominant bending modal

period, respectively, and about 1/12 and 1/6 of the dominant torsional modal

period (refer to Table 1). Step sizes greater than 5 seconds will be

undesirable due to the increased modulation effect on the output. Tile

improvement of 2.5 seconds over 5 seconds has not been significant. Note

that the solution algorithm can tolerate much greater step sizes than

the numerical integration algorithm. For step functions, exact solutions

will be obtained regardless of the step size used.

3.3.3 The Simulation Outputs

The outputs of a simulation run are plotted as time histories

of all the interested quantities. The plot consists of the modal amplitude

of all the flexible modes, the kinetic and potential energy of the indivi-

dual flexible modes, the sum of the flexible modes, the rigid body, and

4

I

b

41

v

the sum of all the modes; (the energy plots are particularly useful forP

identifying dominant modes); the maximum out of plaice displacement; the

three kinds of local slope angles, i.e., the root-mean-square spatial

average (RMSL), the root-mean-square running average (RMST), and the

maximum slope (SLOP). The antenna scan losses due to slope deviutiors

from the line-of-sight were computed and plotted in two ways, the first,

labeled as SLOSS, the scan loss of cacti of the 61 surfaces computed

individually, and their effect summed; the second, SLRMS, computed by

using the RMS average of the entire array. Although these are distinct

terms, their values are practically the same.

In addition to the information plotted, in the output

listings the surface of the maximum slope and the grid point of maximum

displacement are listed at each time step.

3.3.4 The Surface Deformation and the Scan Loss

A. Surface Flatness Subject to Collector bending oscillation

The effects of collector oscillation of an amplitude of

1/2000 of its length, or 10 m, are discussed here. From Eq. (38), with

z = 10 nt, ma R 15 x 106 ks^, ws R .00675 rad/sec, the amplitude of the

acceleration force, Fm will be 6832.8 N, or the forced acceleration is

.0465 x 10- 3 S. Figure 14 shows the responses of the structure. By

examining the modal energy (Fig. 14 (e)-(h)), it is quite clear that the

first three flexible modes (7,8,9) are dominating. The oscillatory

property of Fig. 14 (a)-(d) shows basically the vibrations of the dominant

modes whose frequencies range from .0189 to .0196 Hz.'^

The surface deformation under this disturbance has exceeded

the allowed limit as the MIS spatial average slope angle has exceeded

0.09° (or 5.4 arc. min.) shortly after the force has been applied and it

still reached about .07° after 930 seconds as shown in Fig. 14(a).

Figure 14(b) shows that the corresponding scan losses of 5.769 and 3.52%

respectively. Based on the DOE document ( 2, p. 27], the antenna /subarray

42

mechanical, alignment of + 3 are minutes was specified; and in Ref. 3,

pp. xiv and 1,3 the mechanical pointing and slope accuracy of 2 are

minutes and 3 arc minutes (during all phases of operation) were stated,

respectively. The scan loss corresponding to a slope error of 3 arc

minutes (or .05°) is 1.79%. Hence the 10 m oscillation is too severe

for the antenna structure.

From Fig. 14(u), the maximum local slope is much greater

than the RMS average; for instance, it is greater than .13° at t = 10 sec.

The maximum out-of-plane displacement is about ± 1 m as shown in

Fig. 14(c) I

It is interesting to note that the energy (envelope) of

the flexible modes leads the energy of the rigid body modes by 1/4 cycle

of the excitation period as indicated in Fig. 14(d).

For the purpose of comparison; two other forcing functions

of the Same amplitude were applied at the antenna interface. Thesefunctions are 6832.8 sin (.00675 t) and 6832.8U(t).

The responses of cosine and sine excitations are drastically

different. Figure 15 shows that the latter is extremely smooth compared

with the former. This is also clear from the energy plot, Fig. 15(d),

where the flexible modes contribute very Little to the system energy.

In contrast, Fig. 14(d) shows a much higher proportion. Again, the

dominant modes are 7, 8, and 9. Figure 15(c) shows that the out-of-plane

displacement has reduced to about half as much and, from Fig. 15(a) and

(b), the local slopes and the scan loss are both staying within the

requirement.

A step function of the same magnitude, has excited theyflexible modes in much the same manner as the cosine :function has excited

them. By comparing Fig. 14(b) and Fig. 16(b), for instance, the two

responses share approximately the same envelope which is slowly decaying

with the time constant of the dominant modes, i.e., approximately 1675

seconds.

43

The results of these three cases are summarized in

Table 4.

It is important to point out that in the contest of controlled

motion, signal shaping is extremely important and so is the thrust cut-off

time (refer to Section 3.3.4D).

B. Surface Res onse to Collector Bending Disturbance due to ThermalDistortion

In A, the main purposes are to explore the effects of signal

types on the flatness of the antenna surface. Isere, two more cases are

discussed relative to the orbital thermal effect. It is discussed inSection 5.2 that the temperature gradient in the collector can build upalmost immediately after the SPS moves out from a full shadow and it

is estimated that for the material with the coefficient of thermal

expansion (a) of 10-6 m/m/°G and a temperature gradient of 100% front -to-

back, the collector bending can reach the amplitude of 6.67 m. With

the first collector bending mode dominant, the corresponding force will

be

F - 4558.5 cos (.00675 t) N

and the corresponding acceleration will have an amplitude of .031 x 10 -3 g.

The simulated results indicate that this disturbance will

cause more local surface warping than specified. From rig. 17(a)-(d)

the maximum UIS slope exceeds 0.06° and the scan loss is about 2.7%.

Therefore, an a of 10 -6 m/m/°C with 100°C gradient is not acceptable.

However, an excitation force of

F - 3417.2 cos (.00675 t)

corresponding to Z = 5 m (or 1/4000 of the length of the collector

structure) will be within the requirement as indicated in Fig. 18(a)-(d),

where the RMS slope error is about .046" and the maximum scan loss is

1.45%. The maximum out-of-plan displacement is 0.5 m. From eq. (34) and

eq. (35), the corresponding a for AT = 100°C is .75 x 10-6 m/m/°C.

S

r

x

Table.. 4. Comparative Performance Analysis, Z. » 10 m.

Applied ForceN 6832,8 coa(.00675t) 6832.9 sin(.00675t) 6832.8 U(t)

Ace. Amp., g .0465 x 1,0-3 .0465 x 10-3 .0465 x 10-3

Sim. Period, sec. 1000 1000 1000

Dominant Modes Modes 7,8,9 Modes 7 0 8,9 Modes 7,8,9

Local Slope .0910 .049° 109050(RMSL max)

Scan. Loss 5.76 1.73 5.72(SLOSS) %

Slope at .0690 - .07056t - 930 sec

Scan Loss at 3.520% 3.36%t a 930 sec

Displacement 1101 .56 1.02

Wz max' m

c

Table 5. Analysis of Performance: Distortions Caused byThermal. Disturbance, a - 10- 6 and .75 x 10" 6 m/m/ "C

4

Coefficient of Thermal. Expansionm/m/°C

Front-to-Back Temp. Grad., °C

Thermal Bending Amplitude, Zm

Ace. Amplitude, 6

Ace. Force, N

Local. Slope (MISL max)

Scan Loss (SLOSS max),

Max. Displacement (Wzmax) , m

10-6 .75 x 10-6

100 0 C 1000C

6.67 5.00

,031 x 10-3 .023 x 10-3

4558.5 cos(.00675t) 3417.2 cos(.00675t)

.06 .0460

2.7 1.45

.7 .5

45

Therefore a value of a less than .75 x 10-6/°C will be acceptable

provided that the temperature gradient will not exceed 100°C. It seems

that an a of 10-7 m/m/*C will provide a good safety margin. These

results are summarized in Table 5.

C. Surface Deformation Subject to Collector Torsional Oscillation

Torsional vibration at the collector boundary can produce

large torque at the antenna interface if the amplitude is

high. In section 3.2 it is estimated that, for V oscillation, the

torque introduced to the antenna will be

T 6,453 x 106 cos(.0226t)

with a corresponding couple at 1000 m apart of

F = 6.453 x 103 cos(.0226t).

Figure 19 shows that the resulting surface distortions Are within the

requirement. The maximum MIS slope is .05 0 and the scan loss is about

1..8%.

The dominant modes in this case are modes 10, 12, and 20 as

indicated in Fig. 19.

For the purpose of comparison, another simulation was made

with

T a 8.397 x 106 sin(.0226t)

or the corresponding couple at 1000 m aparw,

F u 8.397 x 103 sin(.0226t)

which is equivalent to an oscillation amplitude of 1-.3 0 . A much 6moother

response was obtained as shown in Fig. 20(a)-(d). The results are

summarized in Table 6. These results again point out the importance of

signal shaping.

46

Table 6. Analysis of Performance; Surface DeformationCaused by Collector Torsional Vibration

Amp. at Boundary 10 1.30

Angular Ace. rad/sec/sec .51 x 10-3cos(.0226t) .66 x 10-3sin(.0226t)

Ace. Torque on Antenna, N-m 6.453 x 106cos(.0226t) 8.397 x 106sin(.0226t)

Force Couple at Interface, N 6.453 x 103cos(.0226t) 8.397 x 103sin(.0226t)

Local Slope (RMSL max) .05° .0420

Scan Loss (SLOSS max), % 1.3 1.2

I

Max Displacement(Wzmax), m .7 .44

Table 7. Comparative Performance Analysis of Antenna underRectangular,-Pulse-Forced Acceleration

Forcing Function F = 4000[U(t)-U(t-500)] T = 2x106[U(t)-U(t-500)]

Distributed Force Fd ,N 1000 1000

Local Slope RMSL max .0530 .017

Scan Loss SLOSS max, % 2.05 .21

Max Displacement WZmax, m .6 .175

4'

47

{

c

D. Comparative Performance Analysis of Antenna. Under Rectanoular-Pulse-Forced Acceleration

Two cases were simulated, in the first case a rectangular

pulse force

F R 4000 [U(t) - U(t-500) )

was applied at the antenna interface and in the second case a rectangular

torque

T 2 x l06 U(t-500)1

was applied to the antenna.

The objectives of this experiment are to make the following

two points;

a. The antenna has greater rotational stiffness than

translational stiffness. This observation may be-

verified by comparing the two cases as plotted in

Fig. 21 and Fig. 22. The force of 4000 N was distributed

over 6 points (refer to Fig. 13a) and the same force

was used to produce the torque of 2 x 10 6 N-m by reversing

one of the two 3-point sets of distributed force (refer

to Fig. 13(b)). Note that the force F in Fig. 13 is

.25 F. The translational acceleration has caused a

RMS slope error of ,053° and .6 m maximum displacement

whereas the rotational acceleration has caused only

an .017° maximum MS slope error and .175 m maximum

displacement. Table 7 shows the comparative results.

b. The cut-oft time of the pulse is important. For instance,

in the case of the translational acceleration, if the

pulse were cut off at about 26 seconds sooner (or later),

the antenna surface would have been flat after the cutoff.

This is quite clear from Fig. 21(e) and (d), since at

those times the flexible mode energy was at the minimum.

t,

..

48

6

3.3.5 Further Discussion

In Section 5.3.4 the analysis is based on simulation, results.

However, no numerical information is by itself complete. In this section,

we shall complement the numerical results with further analysis.

Let q be the modal amplitude of a dominant mode, and the

corresponding differential equation with tine excitation of cosine function

may be written in the following Corm,

q + 2 ,wii + w 2 q h cos wo t

(39)

tine solution of this equation with zero initial conditions is,

(w _w 2 )h a twt C(w2^

q(t) 2 °2 2 (- cosw 31- 't , - 2 2

2)sin(4/_ 2t)

(w -wo )+(2rw wo ) (w -wo ) l-r

(w2-wo2)h 2Cw w

Z+ ^2 2 2

(cos wot + 2 Bile wo t) (40)(W (d) +(4w wo) w -wo

In the context of this application, , is in tine neighborhood of .005 and

wog << w2 , (40) may be approximated by tlae fallowing simplified equation

q(t)21 2 {cos w 0 - e-cwt cos wt} (41)

W -w0

The first term inside tine braces represents the steady state solution and

the second terra contributes only to the transient response.

'file maximum value of y(t) is 1.983h/(w 2 -•w02) which occurs att = 26.36 sec. Th e transient to the steady state envelope ratio is

about 2 to 1.

Since the modes are decoupled in our model, (41) may be extendedto all the flexible modes; Bence for mode k, k = 7, ..., 20,

hk

yk(t) 2 2 {cos w0 t — e- wkt cos y) (42)Wk -wo

From (42), we have the following observations:

1. The modal response is proportional to the constant hk.

Since h is the product of the inverse of the kth element

of the generalized mass, the mode shape at the point

where the force is applied, and the amplitude of the applied

force, the modal response is proportional to these

parameters.

2. For w k 2 » Wo2 , which is the case in this application,the modal response is inversely porportional to the square

of the natural frequency,

3. Since the surface deformation at a given point is the sum

of the products of the local mode shape and the corresponding

modal response, the above properties apply directly to the

surface deformation.

4. Owing to the fact that in this application the local

slopes are so small that the slope angles are proportional

to the local deformations:

a. For a known system response to a specific disturbance

level, the responses to other levels may be obtained

by linearly extending the known results.

b. For known responses to various types of excitations,

new results maybe obtained by taking the linear

combination of the known results.

By application of these linear system properties, repeated

experiments may be avoided and hence a great deal of time

and effort may be saved. Figures 23, 24, and 25 show

the performance characteristics of the antenna over wide

parameter ranges. These figures were obtained by applying i

the above linear properties.

50

k

Table 8. Key to Variable Names for Figures lei to 22

EFLSX The sum of energy for the first 14 flexible modes, Joules,

EL7-1L20 The modal energy for geodes 7 to 20, Joules.

BRIGID The sum of energy for the 6 rigid body modes, Joules.

CSUM The sum. of the flexible and the rigid body modal energy,Joules.

RMSL The root-mean-square spatial average of local slope angles,degrees.

RMST The root-mean-square time average of local slope angles,degrees.

SLOP The maximum local slope angle at time T, degrees.

SLIMS The EMS scan loss (computed from RMIM), percent.

SLOSS The scan loss of the array, percent.

T Simulated time, seconds

WZDIAX Maximum out-of-plane displacement of the array at time T,meters.

YL7-YL20 The modal amplitude for modes 7 to 20.

51

(U)

ANG 0EV IN GEG-RM5 RUNNING AVE,RMU SPATIAL AVE,MAX

m

p

n t

M= RMST i (a))0w RMSL d

I

A = SLOP

i

n . ZOn +o(t• 600 000 Ma. Moo Hou• 1 OT

PERCENT SCAN LOS5 AND RMS SCRN L055

w

w`.

m

N

a: $LOSSb

^^ SLRMfi

M

ry

k

0. too • 400, boo Don- )OM 1200- 1400 :tlW,

T

Figure 14. Surface Response to Collector Bending OscillationZm = lom, F : 6832.8 cos (.006751).

52

î's4

MAX OUT-OF-PLANE DISPLACEMENT

NZMAX

ZOP •+00. M). 000• loco. 1200• 1400 160ET

KINETIC AND POTENTIAL ENERGY

(d)

xt

M= F.SUM

Y " FRIO10

a_ MEX

i

0- Z00• 400, 400. 000• loco 120o. 1400 )000T

Figure 14. Surface Response to Collector Bending Oscillation:Z = 10m, F = 6832.8 cos (.00675t)(Cont.).

53

i

Mz EL7

ELO

EL9

ELIO

of

ti

(e.)

KINETIC AND POTENIIAL ENER5Y

0. M. 400. 60• 000 1000• Itoo 1400• 160D•T

KINETIC ANO POTLN'JAI ENERGY

M-- ELI I

ti

ELI 49

w

t;

t;

0. too. 400• 600 000• 1000 legs 1400 1600T

Figure 14. Surface Response to Collector Bending Oscillation:Z m = 10m, F = 6832.8 cos (.00675t)(Cont.).

*54

x

r

I

is

^l

(g)

.0

t0eELIS ^?

Mz ELI 60

A- EL17

vt-ELO

!.

(k1)

Q

KINETIC ANO POGL WIAL ENERGY

0• too. 400. 000• Da0• 10¢0• IZ00 1490, 160¢.

T

KINETIC ANO POTENTIAL ENERGY

V

M

nri

N

04 EL 16n

EL2ON

V?

n

N

a. zoo 400. 090• 000 1COO. 11.100 1400 1000

T

Figure 14. Surface Response to Collector Bending Oscillation:Zm = lom, F = .6832.8 cos (.00675t)(Cont,).

x

4

55

(D= YL?

r. YLSYL9

(1)

MOVAL COUROINATV,

0^ ZOO• 400• too ON loco. KOO 1400 loco

7

MOM, CODROINPIES

a)-, YL 10

YLI I

&z YL 12

FbbI

Vp

O

0. ZPG - 400 600 two 1000 1200 140c, 1F10

7

Figure 14. Surface Response to Collector Bending Oscillation:Z m 10m, F 6832.8 cos (.00675t)(Cont.).

56

(1♦

C

0 r Y L 17

Wa YL 18

A t YLIS

st v YL20

tij,"At

—4-

Cs M ^ ^^ ^^ l.^â'1^SLS.'11^x"'Pw' ,. w_ _ .._.__. _; ê-^rr..:«a^, ^_.:.X=YL14

YL I SJT

Yi. 16

01 coo- 400. 020 rea 1010 IrM 1400, locc.

MOUPI U.IRDJNAIL f

0. ZOO 400• 000, 000. 403, WC- 1400• 160V

T

Figure 14. Surface Response to Collector Bending Oscillation:zm m 10mv F - 6832.8 cos (.00675t) (Cont.).

57

ANG DEV th DJ.(#'-MH5 RCNN:1,,; PVE,IM I) Rvf,.HRIA

M-» RM51

XORMSL

a, 900 . 400 000

E00 1000. IM I 0^0 1000

T

PERCENT SCAN LM AND RM5 SCAN 055

fi

C

(b)

ri

SLOSS

SLRMS

04

a tW 400 ooc- 00 1000 IM, 1400• 1000

T

Figure 15. Surface Response to Collector Bending Oscillation:zm = 10m, F = 6832.8 sin (.006751).

(a)

A

58

! Y

BG, h

t

?ice ... _`^-..r.-3P^..'^f,â^^.. ,.. t ... «.;• .^y"^^^.b%',^*.°^. • -.. _..va^4. t .,..

0

v

N

J!

WZMAX

t

R

(G)

ahs E^1C^J;<t

L : EP'tEX

+5 n

(d)

MAX OUT- OF-PI,ANB DISPLACUNLINT

M

t0 coo W Gw0 4«"w IOC., I*= 14t IM,

T

KIhETIP RNtI PLATEN*IR ENCRUY

o OU0 400 000. G." 10 1ZVI 1+0'; IOZOT

Figure 15. Surface Response to Collector Bending Oscillation:Zm = 10m, F a 6832.8 sin (.006750 (Cont.).

59

I

F

aw EL? ixm ELQ

A t [L9

KmEL1Q °g

i

a

(0).

MN

Vi

NN

I'1

ry

V1

n

0:ELII

EL 12(Y

^^ EL13 ..p

x EL 14

n

C7

N

Mh

C

(`)

*

KINETIC Rh.11 P3tEh?jkL ENUNY

4 200 400 000 CGO Wcoo 1"00 140. 1 INCT

E[ 0 too 400 ON Eoc^ loco 1coo 1400 loco

T

Figure 15. Surface Response to Collector Bending Oscillation:

Z = 10m, F 6832.8 sin (.00675t) (Coat.).

60

1 wMCxsf^+uvr'-s:.w .:.. 4 .« --t x..!. I-e_x, +ems ,x:...

..... ... .L. ,3s ^.. _. .... ..:.. -x vim,. .. •.,5.^.3?Y^....51^k»'-^J;S.a^)e^.,_a.,...^..`.,.. ..,. Y ..?r=ii.F_ ^ ,.^..^-£-_.-.^^v ,-_.«a.

(h)

O=ELIS

X= EL20


(g)c7wELI5

xA ELI

ELI 7

m=ELI610

0. 900. 400- 600, BOO. 1000. RM 1400• 10•T


M. 400, 000. too. 1000, 1200. ma. 1000,T

Figure 15. Surface Response to Collector Bending Oscillation:zm 10m, F 6832.8 sin (.00675t) (Cont.).

61

1

MOORL COE RPINRTE S

Clc YO

K. YL9 ? ^^ _ W._,

• YL9

i

0 Coo 100w too. 000 1000 1200• 1100- ON

T

HgE)RL ^C10ftU1NPTÊi

Ya

a

M YL1Q

x,YL11 R._0

-%aYL12

ft

0 C00• 100• 60C too- I00c, lion- lA= 100C

T

I

Figure 15. Surface Response to Collector Bending Oscillationt i

Z m _ 10m, F 6832.8 sin (.00675t)(Cont.). 1

62

a

P

1 n

MODAL COOROINRTE5

A

N

i

T-FftA

O

0: YL13 (k)

Y 14

p^ YL15

K=YLI6o __

0a

1

H

1

ryn...., a,

1400, 400. 000 Boa. 1000 Woo 1+o0. 1000T

MODAL COOROINATe$

e8

108

B

4_ YLI7

(1)

x- YLI6

n=YL19

e= YL20 On

Y

L

î

PI n

t !

8

Figure 15, Surface Response to Collector Bending Oscillation

z lom, F = 6832.8 sin (.00675t) (Cont.).

63

• .s

3. !

r

Tl ^,

HM

N

b

A. RMST

w-,RMSL t

A- SLOP

fJ

(a)

RNG UEV IN OW-RMS Rt:NNINU RVEMS SPATtAL AVE,MAX

C r0'3 400 600 000• 1000• 1200• 140O• 1600

T

PERCENT SCAN LOSS AND RM5 SCgN LOSS

F 1

Iul

C

(b)

^te:iLRM5

v

a

y

0• "0• 400• 600. 1100. 1000• 1200• 1400 1600

T

Figure 16. Surface Response to Step Acceleration, F = 6832.8 U(t).

a ^

t.

64

A

MAX OUT-OF-PLANE DISPLAMIENT

a

(c)

HUM 4

N

0

NCO• 400• 6Cq• 000• 1000• 1200• 1400• 1600•T

KINETIC ANO PQTENTIRL ENERGY

n

C1: E50M

x=LMI0

s = CrLEX g

F,nv

I

(d)

t90

AU

4

Z00• 400• 600• DO- 1000. 1200• 1400• 1600T

Figure 16. Surface Response to Step Acceleration, F = 6832.8 U(t)(Cont.).

65

~

^

Q

^

G

^M.- RdoT

w=RHSL

SLOPC?

^

C,

^

Du un 'too' mm vm' mW NM nm mm^

T

e"

U7~

C?m

~*

m"8L8GS

W^t SLnMs ~

~

^

r

q. —. `_ °~. ,~ °~. ~°. 700 ~~^

. .

(6)

Figure I7 ° Surface Response to Collector Thermal Bending Disturbance,m "~ I0-6 m/m/"C, F = 4558.5 cos (.00875t)~

86

d

m

NZHRY, H

4

N

at

m

w0. 100• e00- 000= 400• 600. M. Mg. vuv-

T

(c)

KINETIC RNO POTENTIAL ENERGY

r

(d)o_ esun

>KÊRIG1]0 8D

^- EFLEX

N

€tn

a

100• L00• 300. 400, 600 600 700- 000

T

i. Figure 17. Surface Response to Collector Thermal Bending Disturbanceg P g ,a = 10-6 m/m/°C, F 4558.5 cos (.00675t) (Cont.).

67,. ii

(a)

to

rJ

O

a^ SLO5S

)S. SLRMS

I?

(b)

A

ANO DEV IN DEG-RMS RUNNING AVEMS SPATIAL AVEoMPX

0 It—it I

8

s.

p.RMST

wv- RMSL

•= SLOP

s

O -

Q

0t 100. too- 500• 100: 600+ 400+ 700• OGO*

T

PERCENT SCAN LOSS RNO RMS SCAN LOSS

.I

0• 100, too- 000. 100• 600. 600• 700. 000,

T

Figure 18. Surface Response to Collector Thermal Bending Disturbance,a = .75 x 10-6 m/m/°C, E = 3417.2 cos (.00675t).

68

MAX OUT OF PLANE DISPLACEMENT

V4 A

(c)

ANznax

k

i

t0I

hi

100• 200, 300• 100, 600• 000• 700, 100,

T


a

m

0r,

a

NV%- ESUM

x;ERIOIO ON

Ac EFLEXQ

nna

x

0ua

i00. 200. 900, 100, 500• 000, you• auu.

T

Figure 18. Surface Response to Collector Thermal Bending Disturbance,a .75 x 10'6 m/m/°C, F = 3417.2 cos (.00675t) (Cont.).

i

69

if

reft J

..._ .._ ._ _ JW .^L .... _.:. _. '.' =c_ vi. ..._ a.` .1.^. 1.^ _,..SaS .__ 1. _..'a.. ......0 :- s ^. 3 r nn..

d)

U. RM31

xc RHSL S.

6LOP

ANO DEV IN OLG=R MI3 RUNNI N G A VEAM5 SPATIA1 fiVEMAX

0. 100• too. 000. 400• $Co. Doc, 10 pecc,T

PER CIEN7 SCAN L055 PNO HM,1 SCAN 05'5,

SLOSS

01 100• too • Ogg 4• 600 600 wc 000

T

Figure 19. Surface Response.to Collector Torsional Oscillation,6 = 1 0 , T = 6.453 x 10 6 cos (.0226t).

(b)

A

70

(d)

M= ESUM

M»ERIoto

a^ EFLEX

â

t. i

MAX OUT-OF-PLANE DISPLACPMNT

x

(c)

A

1

1i

iVim__.

U . 100. too. OW 400. 600 000• 7D4 UCD.T

KINETIC ANo POTENTIAL ENERGY

t;_

l

0• 100• 400 oca. 400. 50, 604 109. PooT

Figure 19. Surface Response to Collector Torsional Oscillation,6 = 1 0 , T = 6.453 x 106 cos (.0226t)(Cont.).

► 71

M-- CL7

ELO

EL9

ELIO

9

(e)

v

E

M=ELI]

x.ELIZ

z=EL14

A

KINETIC Win Pf1TfN'jAj- CNEF6Y

V. 100• too. 000• 400• fool too, log- PooT

KINETIC AND POTENTIAL ENMY

Al

fj0. IM log. too-

Figure 19. Surface Response to Collectvr Torsional Oscillation,1 0 , T 6.453 x 106 cos (.0226t)(Cont.).

72

e

.

(D)

,fix

KWIN AND 11 07001k ENEROY

k

S I ^ 9

lF k

Clm EL 16

1t Et^10 ^ _

^ M

A =ELI 7

0. 100 to 094• 400, W 6% 190 000T

KINETIC RNG POTWIPL E,NERVY

2Y

3 r' ..

ANtmza^f"RRw.'. 'a.ac.-asxr rr+r^^ _>s -a=..0 ..sa m ..: u,. ^ 1

„ t

^2

Y

^^ 2

tq-^Llg

X- EL20

" f !1^ .^q. ivq . zcq. aao= âq ^

5^p• eqa.^ âq ooa.

T

Figure 19. Surface Response to Collector Torsional Oscillation,0 = 1% T - 6.453 x 10 6 cos (.0226t) (Coat.) .

73

If

r^

p : YLIt1

x4YLII t°

N► _ YL 12

4

el l

"004 [;EMI NATEb

M.- YO

>« YL8y

a YL9

a

' p. 1p0• Lao• Dao• 101 Wo ovo 700 oto

7

t93"M COMINAILb

t rk,r c

Q . loo. Loo No. +aa• AUDI no IN- DDar

T

Figure 19. Surface Response to Collector Torsional Oscillation,8 - 1% x _ 6.453 x 10 6 cos (.0226t)(Cont.).

74^

a

6

0

n0

N0 (1)

MODAL COORDINATefi

i

9

Mn

W;; YL13

w=YL148

&=YLiS

A =YL160

S

S

JO

it1

(k)

MODAL COORDINRTe3

M= YL17 a

*=YL1B

=YL19

g = YL20 p

a

v1

0

i

0• 100• Z00. 500• 100, 600• 400• 700. 000•T

Figure 19. Surface Response to Collector Torsional Oscillation,6 = 1% T 6.453 x 106 cos (.0226t)(Cont.).

75

h.

.A'06

4

r

Isa L-

PF,RL[.N' ^.AN LC"o" SN" P'.'^ 5k;'Q

AN• W, IN OUi-Rt RL kvNIN ,v AvL (V^F A^r, '.A

P

C. IC1.1• coo N1.T

(a)

(b)

1.__I1

I

T

Figure 20. Surface Response to Collector Torsional OscillationO m

1.3', T 8.327 x 106 sin (.0226t).

76

71IT1IIt

(c)ce

wzrsax

r

MAX OUT-OF-PLANE DIMACEMENTW,

C. 100, Na. "IN A 1: • 501 vq. u-. -. •T

KINTTIf FIN^^ M:r^—ov

FFLEX

Sy

0

scar LQa' 30- 400, 51N• U01— 70-' . K.U.

T

Figure 20. Surface Response to Collector Torsional Oscillation,6m1.3% 8.327 x 106 sin (.02261) (Cont.).

(d)

77

PERCENT SCAN LULS ANO RNr SCAN Lp^S

n

SA1 i

,

1 j

I

i

i

3

E

n^

t.

t^

G1

W

SLOSS

x SLRHSn

19

(b)

ANO OEV IN t)EG-RK9 RUNNING AYEAKS S PA"tAL AVE,MPX

a

F

(a)

p _ RMST

0(-- RNSL 4. e _..

•° $LOP

o• IOa• zoo. 300, 400• 600 600. Vol Soo,

T

0. 100• 200. 300• +90 500- EM 10: 3=

T

Figure .21. Surface Response to Rectangular Pulse Acceleration - Translation,F = 4000 W(t)-U(t-500) }.,

,73

{ ii

raw—•.. v-.....e*—.sxr^mcec^.uz'uc;êyr

r

00QD

0

d ^-

MAX OUT—QF-PLANE DISPLACEMENT^S

s

v

N(c)

wzn^x

t

0. 100. 2p0, 300. 4W 000. 600• 100• 000•

T


nDe E6UH o

IK:ERCG[D

m=EPLEY c4a0N

bqnN

C}nRC

(d)

100• P.M. 30d• ^pU• E00 600. 100 000•

Figure 21. Surface Response to Rectangular Pulse Acceleration - Translation,F 4000 {U(t)—U(t-500)}(Cons,).

79

a, EL?

w= ELO

J% E L9

Y=EL10

i


(e)

0. 100• 200• 300, 100• 600• 600• 700. 000•

T


•

R

d-EL1I (f)

w»EL 12

mÊL13

X=ELl4u+

o

00 - 1001 200• 300• 400• fi00. 600. 700• 000

T

Figure 21. Surface Response to Rectangular Pulse Acceleration •- Translation,F = 4000 { U(t)-U(t-500)) (Cont.).

80

P

p

Qs [1.15 ri

w= Et.16

•=EL17p

KG E41 V 17

ri

A

1 ^ j

1

U H

11'^

Y 1 ^.3^ ^ ^

f i^^

1

t^ ^,Ii ';

'^ f" r y i a

I ^^,ÌIIIIIÌT

I Ica iN7

.^

-...

7

A

(g)

w

N

p

o=ELI 9

^t EL20

Iq

P'

(h)


n• 100• 200• 300• 400• 800• 800• 700 900.

T


0• 100• 200. J00• 400, 800. 900. 700• 900.

T a

i

i

Figure 21. Surface Response to Rectangular Pulse Acceleration - Translation,F 4000 (U(t)-U(t-500)} (Cont.).

81

'Ak

1 ^

kk^

^

YL 10

YL I I

&=YL12 9

0 1

^ j

32

HOOK. CUORONATC5

M

M= YO

M;; YLO

YL9

Q. too. NO. NO- 400, toe. 600- 700, 1001

900AL f=RDTNRTE5

91

0 •100. 200, 300• 001 GIO, 600 100. 500•T

Figure 21. Surface Response to Rectangular Pulse Acceleration - Translation,F - 40GO {U(t)-U(t-500)1 (Cont.).

a

u

aOx YL13 99

X-- YL14

A-,YL16

x_YL16 89

P

1

vtl

J .,

1

MODAL COVIROINATFS.

v1

l

1

Y

^^Tn'.i^11(Jr

1 }^^^ l^f ^f^rîi

^Y1

`^^ Y^1

CV

M= YL17

W:-YL18

A = YL19 u

s^ YL20U

Nv

i

p0

s

N00AL COUROTNATEV

0. 100. 200• 300. 400. 600 800. 700• c00T

(1)

0 - M. 200. 300• 400 601 E70 700. 9C0

T

Figure 21. Surface Response to Rectangular Pulse Acceleration - Translation,F 4000 (U(t)-U(t-500)r (Cont.).

fi

83

b)

ANGi OCV IN pW M, RUNWN5 AUK ? N^ SrÊ7M11, AVr ' . px

P y

M RNST1

(a)

W--RNSL

A t SLOP

p

N

0. 100. F00> 300. 400. %0• 900. 7C0. 0

T

PERCENT SCAN LOSS AND RHS SCAN LOSE

e

>;1

n

NA

G^

SMS

X= SLRMS

c^

u

6A^f

0^ 100. .100 300. 400. M. 600. 700. 000.

T

Figure 22. Surface Response to Rectangular Pulse Acceleration - Rotation,T = 2 x 106 {U(t)-U(t-500)}.

A I

84

MAX OUT-OF-PLANE UISPLACMIENT

4

(c)

v

NZHflX G

a

G

a

N

0 100. too. 000. 500• [Go boo '+00. 00G•T


4

rÊ;

(d)

[7.;. E uUH

xr ERtliCi7

AzCr'LEX ut3

,3k1t)

a

100. too, 300. 500. wo• @0e• Wo. N.G.T

Figure 22. Surface Response to Rectangular Pulse Acceleration - Rotation,T = 2 x 106 (U(t)-U(t-540)} (Cont.).

854.

(e)


r

r

tic. CLx7

K.e^fl

^CL.S

ELIO

Î

C• ICQ. :00. a00^ 140, ..

6G0^ +^r. 1CC^ :l00

"t


0W

c7ÊE.tt ^

K^4L<t2qqa'

A, Et_t.

K=EL 14 ri

9

100. 200• 300• 100, 400. 600, 700• W.

T

Figure 22, Surface Response to Rectangular Pulse Accel p.ration - Rotation,T - 2 x 106 {U(t)-U(t-500)) (Cont.).

86

r

k

4

t


19

n

rin

M; ELI 5 it

Mr,

^ tLkT r•

XzELIL r.

4)

0 IM No- 101" 4; w" IGO• 4+0' V0. !OYi


^x

mcE 1„[9

>«:. n yy

0- i00• 20Q• 100. 400. 600• ""=v' E00

T

(8)

(h)

Figure 22. Surface Response to Rectangular Pulse Acceleration - Rotation,T - 2 x 106 {U(t) -U(t-500)} (Cont.).

87

i

^t

I

MODAL COOROCNATCS

M.- YL7

M. YLO

N. YL9

v• Ûp• tPP• i00• iPA^ 6G0- 6Cd- 100• swP

r

Mount, CODROINATCE

,^ 4

(7: YLIOr J

)Kz YU I

•ZYL12

YH

Y

y^pp

f

t

if

i

i 0• ... 100• ¢00• X00• SPA• GGO. BP0• 70P. 000

T

ti iJ

Figure 22. Surface Response to Rectangular Pulse Acceleration - Rotation,T - 2 x 106 (U(t)_U(t-500)) (Cont.).

88

-,,-

rG

++I

1

Mz YL13Cf

Xz YL14

•=YL154

xz YL16 gA

S

N8

TA4

^• 1119 zoo• 900 BOO• 509• 590• 700• ouv•

7

MODAL COOK MPTES

M

M^, X

01

1

(k)

I1

1

! lrrl ttyy Va i

a

n

C7aYL17

X:: YL18

n=YL18a

YL2O

nIt

b

0CJ

l

(l)

MODAL MRCINUCG

Q. too. zoo. 1(1 400• GOC• 500. 709. M.

y

s

Figure 22. Surface Response to Rectangular Pulse Acceleration - Rotation, 5T 2 x 106 {U(t)-U(t-500)} (Cont.). I

89

.09

vrn

v

Eq .08

ZQ

.070V)

tO0 .06

a:.05

O^ V

w

6

N0

4 Z

U

3

5

2 9

AMPLITUDE OF ACCELERATION FORCE, N

1367 2733 4101 5468 6834 8201.10 7

Ê .04

FDOM ` ' 019 Hz

.031 1/ 1 1 I I I 1 I 1 1 0

2 4 6 8 10 12

OSCILLATION AMPLITUDE AT :A NTENNA INTERFACE, M

Figure 23. MPTS Surface Deformation vs. Amplitude ofCollector Bending Oscillation

90

IR .09mrnar

J .08OZ

O .07

p .06J

.05

.04

6

5V)LO

4 QUV)

3

2 9

AMPLITUDE OF ACCELERATION TORQUE, 109N-m

0.0 16.8 33.6 50.4 67.3 84.1.10 7

a

.03 Lf"" I/ L_ l I I 00.0 0.5 1.0 1.5 2.0 2.5

OSCILLATION AMPLITUDE AT ANTENNA INTERFACE, M

Figure 24. MPTS Surface Deformation vs. Amplitude ofCollector Torsional Oscillation

J,

91

30.0

20.0

E 1, 0001 1 .0- -20, 000 n^------^ xnm

u 10.0aWZ

azz 3.0LUf-

Q 2.0

Q

Z1.0

wUgCL; 0.50

0.3

0.2

ocgI

z 0 c;

IS9oo u

f^

1f

.03 .05 .10 .20 30 .50 1.00

COEFFICIENT OF THERMAL EXPANSION, 10 -6 m/m/ °C

Figure 25. Collector/Antenna Boundary Displacement vs.Coefficient of Thermal Expansion of SolarCollector Structure

I A I

4

92

REFERENCES

1. S.J. Wang, "Dynamic and Control Analysis of the MPTS Antenna, 1Dynamic Model," JPL Eng. Memo 347-56, Feb. 7, 1980, (JPL internal document).

2. "Satellite Power System, Concept. Development: and Evaluation Program,Reference System Report~," DOE/ER-0023, DOE Office of Energy Research,Washington, D.C. 20545 and the National Aeronautics and Space Adminis-tration, October 1978 (Published Jan. 1979).

3. "Achievable flatness in a Large Microwave Power Antenna Study,"# Final, Report, Rept. No. CASD-NAS-78-011, General Dynamics, Convair

Division, August 18, 1978.

4. F.J. Stebbins, NASTRAN Program for MPTS Antenna, JSC., August 10, 1979.

5. F.J. Stebbins, "A Study of the Normal Frequencies vis-a-vis theStructural Parameters of the MPTS/SPS," NASA JSC Memo., September 1979.

f

i

;r

A

N i

93

E1

APPENDIX A

Local Slope and Scan boss Program

eiRUNP/H SLLi3boJOC2(jLrSOLAR9592UO,4999 tol9b/658aASG ► A SPS.i)DELETE ► C 28.i1CATvP 28*aIASGoA 20.&FOR r I S SUb l

SUBROUTINE GRSTR (Y ► NN )PARAMETER NPLOT=8REAL Y(20)vYPLOT(NPL0Tr20)COMMON/GRSTRI/YPLOT

.SJW

CC HERE ONLY THL FLEX MOLES ARE RECUROLU. TO INLLUGE RIvIL OCUYC MODES ► CHANGE THE RANGE OF K IN THE NEXT INSTkULTION TO 1.40C

UO 20 K=792020 YPLOT(NN ► K)= Y(K)

RETURNENTRY GRRCALL MOUT( YPLOT ► 8 ► B ► E(1 ► 3i3HOA=)00 30 1=19NPLOTNRITE(28940) (YPLOT(19J) ► J =1v1O)

30 CONTINUE:00 35 1 =1 ► NPLOTWRITL(28940) 4YPLOT(1 ► J) ► J=11.24)

35 CONTINUE40 FORMAT( WEI2.6)

ENDFI.LE 28REWIND 2bRETURNEND

c1FOR ► IS SUb2SUBROUTINE GE;TOAT(RX•RYtkZPTX#TYtTZr IPES ► ILRUIPARAMETER S NP=166 ► NM=20 oNk=61REAL RX(NP)•RY(NP) ► k7(NP)REAL TX(NP ► NMI ► TY(NP ► NMI ► TZ(NP ► NM)INTEGER IGRD(NP) ► lPL(NL ► 4) ► IFLS(NL94)

CC NP=No of NODESC NM=NO OF MODESC NE=NO OF PLATES ELEMLNYSCC RX ► RY vRZ=COOkD OF GRID POINTSC UX+UY ► UZ=DEFORMATION OF THE GRID POINTSC TX ► TY ► TZ=EIIiEN VECTOR MATRICLSC IGRD=ARRAY OF GRID POINT IDC IPE=THE MATRIX OF THL PLAIL LLLMLNTSC IPES=T.HE SEO NO ARRAY FOR THL URID POINTS IN IPECC READ GRID POINT FROM SPS.MPTbPC

94

Appendix A--Local Slaps and Scan Los e Program

00 10 I =1 r4READ ( 5 ► 11vERR = 10) RX 4 11vkY ( I)vRZ(I)

10 CONTINUEDO 2U I I#NP

READ ( 5i11) I6k 0 (1)vRX ( 1)•RY(1)vRC(I)11 FORMAT(1593FI5.4)20 CONTINUE

CC READ EIGENVELTQRS FROM SPS . MPTEVTC

DO 40 K=1•NM00 30 I=19 3

a READ ( 5r31oERR- 30) TX(IvK)#TY(IvK)vTZ(IvK)30 CONTINUE31 FORMAT(8X9E12.7v2E15.7)

UO 40 I : I o NP

REAO(5.3I) TX(I+K)rTY(I.K)#TZ(IoK)40 CONTINUE

CC READ PLATE ILLPENTS ^KOM CARDSC

READ ( 5 ► 51) ( ( IPE(1vJ ) tJ_1r4 ) v1=19NE)51 FORMAT(1615)

CC TESTC

WRITE ( 6R51) ((IPE ( I•J)vJ=1t4) • I=1gVE)CC DETERMINE; SEO NO FOR IPEC

UO 6U I=I.PNEDO 60 K=11NPIPES ( Iv1)=IPE(I.1)DO 60 J=2#4IF(IPE ( I*J).LQ . IGRUfh)) IP£S(I•J)=K

60 CONTINUECC TESTC

WRITE ( or5l) ( ( IPES ( I•J)rJ = 1o4)v1=1vhL)RETURNEND

@FOR ► IS SU83SUBROUTINE MSSE ( IPE.S P IN E vUXPUYrUZ • ICNT 9RMSTPvC1vA8C1#SLOP P ISLOP0

1 RMSLvRMST*';LOSS•SLR4S )CC THIS SUER COMPUTLS THE LOCAL SLOPLS OR THE DEVIATIONS OF LOCALC NORMAT FROM ITS UNDISTURBED NORMAL VALUE. SINCE THL ANuLES ARLC SMALL • DOUBLE PRECISION FOR SOME VARIABLLS IS RLOUREUC

95

Appendix A--Local Slope and Scan Lose PxoBram

C IPES-THE SEG NO ARRAY FOR THt. GRID POINTS OF THL PLATE EL UNTSC NE=NO OF PLATE. ELEMLNTS IN ThIS COMPUTATIONC UX;tUY PUZ=THE INSTANTANUOUS COORDINATE. VECTORS OF THE. GRIDSC SLOPZMAX SLOPE IN THL INTERVALC ISLOPcPLATE ELEMENT NO OF MAX SLOPEC RMSL = SPATIAL AVERAGLv RMS SLOPE, ERROR • ULGREEC RMST=RUNNING AVERAULt RMS SLOPE ERRORv OLGREE

DIMENSION IPLS( 61x4 ) #UX(166) • UY(166_)9 UZ(166)DOUBLE PRKCISION A.b9LvCC•A84INTEGER H

CC PDL -PI*0L/LAMbGA *( PI/18O .) 9 THETA IN ULGkLEC DE=10 . 39 METLRSC LAMDA- • 1224 METER 4 FkLU = i.45 bHZ)C SLOSS IS THE PLRCLNT SCAN LOSS OF THL ARRAYC SLRMS IS THE SCAN LOSS COMPUTED bASLu ON RMS SCAN ANGLkC

PDL= 4 . 654375SLOSS=O.

CSLOP=O.RMSL'O.GO SU IE=1rNLH=IPES(IE92)J--IPES ( I,E:.3 )K—IPE5 ( IErr4)

COMMENT HeJv K ARL THL IST9 2NOP 3Rb POINT THkT ULFI NLS PLATE LLCM IEA1_UX(J)—UX(H)A2=UY(J)-UY(h)A3=UZ(J)-UZ(h)61=UX(K) -UX(H)G2_UY(K)-UY(h)t33=UZ I )-UZ(H)A= A3*B2+A2*b3ti=A3Ê31-A1 a(33C=-A2*81+A1*b2CC=DABS(C)ABC-US0RT(A*A+H*b4C+C)

CC COMPUTE LOCAL SLOPL ANGLL IN UEGRLESC

TEMP-SNGL( ( DACOS(CC / AHC))+57 . 2957t$UL)IF(TEMP.LE.SLOP) UO TO 70SLOPETEMPISLOP=IPLS(IL91)C1=SNGL(C)ABC1 _ SNGL( ABC)

70 CONTINUE

a

0

96

i

Appendix A--Local Slope and Scan Laaa Program

C

3

r

C COMPUTE, LLEMLNT PATTERNv RECTANGULAR APERTURL ANU UNIFOpMC ILLUNINATIONC

IF (TEMP . LO ,0) GOTO bU

ANG=PDL*TEMPTEMPI=SIN(ANG)/ANGSLOSS=SLOBS +TEMPI*7LPPIRMSL=RMSL+TEMP+TLMP

8U CONTINUESLOSS= ( 1.—SLOBS /NE)-*10O.IF (ICNT . LO.0) SLOSSZO*

RMSL=RMSL/NE;RMST=RKSTP;RMSTP*ICNT+RMSL

COMMENT RMSL IS THE RMS SLOPE - SPATIAL ANERAGLe kMST IS ThE TIML AVLICNTP=ICNT+I

CRMST =SGRT (RMST/ICNTPIRMSL =SGRT(RMSL)

CC COMPUTE. RMS SCAN LOSSC

ANG=PDL*kMSL

TEMP I -S1N (AN G) / ANGSLRMS= (1.0-TEMP1*TCMP1)*10J.RETURNEND

aASGtA CSSL*TRAN.v1XOT•G CSSL *TRAN.CSSLPROGRAM SLOPE OF 1000-METER SPS ANTENNAINITIAL

ARRAY W(20).Y ( 20)oYLt ( 2G)rYP ( t(j)•YGP (2())•Z(20) • WU(20 ) oRNO(20 ) PL(2^s)ARRAY U ( 20)PbLTA(20)oC8 ( 2O$oLlt2(j ) pSl(20 ) PTPLOT(d)ARRAY RX(166)•RY(16b)#RZ(166)rTX(166.20)•TY(166,20)91Z(166r2)1ARRAY UX(166)rUY(166)*UZ(166)rUU(3)•VV(3)•WW(3)ARRAY WX(166)rWY(166)vWZ11661

ARRAY MS(20)vPF(20)INTEGER KK • NNrNPLOTvIP • NL•ICNTINTEGER IGRD(166)vIPLSt 61.4) 9 ISLOPPIPMAXUT=2.5TF: 750.

COMMENT' THE MAX NO OF PLATL ELEMENTS 1S 61NE =61

COMMENT APPLIED FORCE IN NLWTONFORCE-854.3

CONSTANT NPLOT'8

CONSTANT W=0.r0.â.t0. r0. ► (► .•.1186b56..1191vSiT^.12.31927^.21552Nr ...

97

Appendix A--LaoAI. Slope and Scan Lose Program

9217U462t * 2166395t.270219br.2t+69lb59 *291U5540.29603389*31326789*31755649.323310dr0324895b

CONSTANT MS=3lbO000.93900000 0 #1480GOOD0o2ti8UOOG,#7180000 * r ...178QOOG.•152OOOO.o161OL00.*215OUOOiol340000. p *,.44 9000U. # 3980000. • 2400000. • 72 5 0000 0 v 5443000 i o ...3000000. vIO20000O.r7120000.t28l0000,93260000.

CONSTANT TPLOT=100.+2009*3009*y00i,9500*96ti0.*7U0.r6Gt1.00 L4 KK=1920

YD (KK)-O.Y(KKI=OOZIKKO=0005

L4.000NTINU'EDO L6 KK`1r20

f C8(KK)=SORT(1,0 - Z(KK)+Z,(KKIII

gETA(KK)= ZIKKI /CB(KK)WO(KK)= W(KK)*C8(KK)CI(KK)= COS(WO(KKI*DTD

! S1 (KK ). SIN(WU(KK 1*(DTIRHO(KK): EXPl-ZlKK)*WlKK)ouT)

L60.CONTINUENN=1SLOP=O.ISLOP_ORMSLcQ.RMST=O*C=O.ADC :0.ICNT=OSLOSS=O.SLRMS--O.

COMMENT CALL GETOAT TO GET DATA FROM SPS FILES ANU FROM CAPUSCALL GETUATtRXYRY•RZ•TXtTYtTL•IPLS*IURDI

COMMENT INITIALIZE 01SPLACLMENT'C

00 L10 IP=19166WX(IP)=0.WY(IP)=0.WZ(IP )=0.

LlO..CONTINUEDO LIS KK=1920PF(KK)=(7Z l369KK)*TZ(53oKKi*.5 *(TZt1409KKI*TZ115btkK)* too

E TZt 58*KK ) ♦ TZ(77rKK I l l /MS (KK)L15..CONTINUEENDDYNAMICCINTERVAL C I;.2.5

IF(T.GT.TF )G0 TO FINIF(DT.NE.ClI GO TO FINDO- L16 IP: 1*166Wx(IP)=O*WY('IP)=O.WZ (IP 1=0.

L16*.CONTINUE00 L18 KK=1920U(KK ) =PF (KK)*FORCE*COS l .uO67a*T

98

Appen0 x A---local Slope and Seen Lose Prosram

LI6..CONTINUEDO L20 KK=1 r20YP(KK)`-Y(KK)YOPIKKIz YDIKK)

L20..CONTINUE00 L24 KK=IokOIF(KK.GT.b) bO TO L23

COMMENT COMPUTE RI(ID BODY MOUES

YIKKI:YPIKK) + YDPIKK)*GT + IUIKKI+0*0T1290)YD (KK) = YDP (KK ) ♦ U(KK I*DTGO TO L24

L239. CONTINUECOMMENT COMPUTE FLEXIBLE MODES

Y(KKI=RHO(KKI*(C1(KK)+ BLTA(KXI*St(KK)1*VPIKK)+ ...RHUIKK) o S11KK)+YUPIK WWOIKKI• ...U(KK!*II.0-RHO(KK)*(CIIKK)*Ui 7A (KKI+SI(KK)1)ItWtKK) *64KK))

Y04KK1--RHO(KKIOW(KK)*(SIIKKIICBIKXI 1 *YP IKK! + ...RHO(KK)*IC11KK)- BEYA(KKI*SI(KKII* YUPlKK)t ..U(KK)*RHO(KK)*S) (KK)/4W(KKIOCO(KK1)

L24., CONTINUECOMMENT COMPUTE GRT'U POINT DISPLACLME,NTCOMMENT *+***COMMENT TO INCLUDE RIGIDbODY ! Q TIO AIJ KK IM L445 SIIOULU START AT I

COMMENT *•***DO L.J KK-7t-_nD0 L25 IP =1 r 166WX41P)-'WX(IP)+Y(KK)*TXIIPtKKrWYIIP l»WY(IP)+Y(KK)*TY(IPrK%)WZ(IP)=WZ(IP) +Y(KK) *TZ(IPtI(K)

L25. CONTINUE

COMMENT GRID POINT POSITIONWMAX=0.TEMP-"Go

IPMAX=ODO L26 IP=I P166UX(IP)=RX(IP)+WX(IP)

UY41P)=RY(IP)+WY(IP)UZ( IP)=RZ(XP)+WZ( IP I

COMMENT COMPUTE MAX OUT-OF-PLAN DISPLACEMENTAB SWZ=ABS(WZIIP))IF(TEMP * GE.AbSWZ) GO TO L26

14ZMAX=WZ(IP)TEMPABS(WZMAX)IPMAX=IGRD(IPI

L26, CONTINUECOMMENT COMPUTE SPATIAL AVE RMS SLOPE ERROR ANU RUNNING AVE RMS SLOPL LFROR

Rt4STP=RMSTCALL MSSL(IPES+ NErUXo-JYPUZPIL NTvRMSTPPC9ABC95LOPoISLOPPRMSL9 ...

RMST#SLOSS.SLRMS)ICNT=ICNT+1IF(NN.GY .NFLOTI GO TO L3UIF(T.LT.TP LOT(NN)1 GO TO L30CALL GRSTR(YvNN1

NN-NN+I

W

99

Appendix A--Local Slope And $can Lose Program

L30- *CONTINUEERIGID =O.EFLEX -O.00 L50 KK=tt,20E(KKI = Y0(XK)*YU(KKI ♦ W1K91*W(1(KI*Y(KK1•Y4XK)E (KK) =.5 *M S f KK) *E (KK )

LSO..CONTINUL.00 L55 KK=1 ► 6E.RIGID ERTGIU +E(KK )

L55.. CONTINUE00 L60 KK=7t20fFLEX-EFLEX+E (KK 1

L60.. CONTINUEESUM =ER1010+LFLEXYL1=Y(1) s YLk=Y(2) S YL3=Y(3) S YL4=Y(4) S YLS=Y(5)VL6=Y ( 6) S YL7=Y ( 7) s YLti=Yfb1 S YL9=Y ( 9) 6 YL10=YIIU) 1YL11=Y(11) 6 YL12=Y(12) s YL13=Y(13) S YL14=Y(14) S YL15=Y(ISIYL16=Y(16) 6 YL 17=Y(17) s YLIS=YJIBI S YL19=Y(1 9) s YL2(j=Y(20)EL1=E(1) $ EL2ZE(2) s EL3=E(3) S LL4»E(4) S ELS =E(S)EL6 =E ( 6) S E;L7 = E(7) S ELt3=E ( b) T EL9=E ( 9) S EL10=E(10)

kL11..E(11) $ EL12=E ( 12) S EL13 = E(13) s EL14=L ( 14) f EL15=E(15)EL16 =E116) S EL17=E(17) S EL18=E(i q ) 6 EL19 =L(19) 6 LL20=L(20)

OUTPUT YL19YL2tYL3 ► YL4 ► YL5 ► YL6 ► YL7 ► YL89YL9 ► YL10#YLII t YL12 ► YL1,3 ► ..YL14PYL1S * YL16 ► YL17 ► YL1otYL19 9 YLtO ►R_M.5T#R1( Sl_ ISLOP ► SL OP:G ► Ai3(; ► SLOSS 9 5LRMS ► IPMAX ► wIMAX#ESUMtLkIG1UoLFLEX

PREPAR YLIPYL2 9 YL3 ► YL4 • YL5vYL6rYL 7+YL8+YL99YLiU ► YLII#YL12 ► YL13 ► «..YL14 9 YL15 ► YL16 9 YL17rYLlbitYL19 ► YL20 9RMSTrRMSL * SLOP ► ESUM9 .«.

ELI ► E L2 9EL3oLL 4 ► LL5 ► LL69LL79LL89L L99ELIO ► F.N.11rEL12oLL139 .«.EL14 9 EL15 * EL16+EL17 9 EL18 « EL14 ► EL2i)oSLOSS ► SLRMS ► WZMAXr ..«ESUMPERIGIprLFLEX

DERIVATIVE GRPYD1=1.0Y1=INTEG ( Yu1 ► 0.0)

ENDENDTERMINALFIN-..CON71NUE

CALL GRRENDEND

i to 1 n ..f^M ^^

v* M A r 17 '!&w;',.

100

„ter

r

5 6022 6047 632313 2023 2043 202221 7043 6023 704429 3043 3022 324218 1062 7040 706146 1063 7041 306233 1001 7961 73 da64 7102 7OB1 710372 3103 3063 310400 4041 0061 6342Bd 2045 2U64 204495 2061 2081 6061

172 *083,6113 6344110 4063 2104 2062117 3101 4121 6101

1 6,321 6+044 6022.15 kO24 2044 20)23 1042 61022 704331 1044 202$ 304340 7061 7044 10624a 3064 3044 3063$7 3042 3061 3016166 7101 7082 710274 bU44 6064 604532 2142 2061 204109 6063 6064 606497 2062 2002 20ol

134 6062 6102 6083112 2064 2100 2063119 2102 2122 2101

Appendix A--Local Slaps and Scan loos Program

4

1

START&ADD SPS.MPTOPAAOD SPS.MPTEVT

1 6024 6045 6025 3 4727 6044 6014

9 2021 2041 6021 It 1722 2042 2021

17 2025 2045 2024 IQ 7044 6024 7043

2$ 3041 6021 .7041 27 3042 2021 3041

3) 3045 2024 3044 30 1063 7044 7004

42 3061 3041 1061 44 3062 3044 3061

$1 7083 7067 74184 53 7042 1064 706'1

59 3063 3062 3062 61 3084 1063 3083

68 7101 30431 7101 70 3102 3001 310,'A

16 6043 6063 64*4 78 00442 606'4 6343

84 2043 2062 2042 d• 2044 3063 3043

91 6062 6063 dual 93 6061 60tls 604499 1063 2083 2002 101 2064 20d4 2343

106 20b1 6101 6002 108 2082 2101 7061113 6102 6123 6103 115 6101 612• bIU2121 1103 2123 2102

FACYORt.751LABEL MODAL COORDINATESGRAPH T ► YL7 ► YL4 ► YL9LABEL MODAL COORDINATESGRAPH T.YLIOoYL1l ► YL12LABEL MODAL COORDINATESGRAPH T YL13rYL14 ► YLI$#YL14LABEL MODAL COORDINATESGRAPH TrYL1T ► YL16•YL19vYL23LABEL KINETIC AND POTENTIAL ENERGYGRAPH Tv L7 ►ELdoEL9 ► EL10LABEL KINETIC AND POTENTIAL ENLRUYGRAPH T•EL11rEL12 ► CLI39EL14LABEL KINETIC AND POTENTIAL ENERGYGRAPN Tr:.L15vEL16rEL17rELldLABEL KINETIC AND FOTENTIAL ENCR6YGRAPH TrEL19tEL20LABEL KINETIC AND POTENTIAL ENERGYU4{APH T#ESUMrEFIGIUtEFLEXLABEL MAX OUT-OF-PLAN DISPLACEMENTGRAPH T+12M4xLABEL ANG OEV IN DEG - RM5 RUNNING AVE.

%Ieo

RPS SPATIAL AVLO 4414GRAPH TrRMST9RMSLv5LOPLAPEL PERCENT SCA1 LOSS ANG 8145 SCAN BOSSGRAPH TrSLOSS•5LRMSSTOPAEOVY 25.0SPS.244E OFiFth

101

AppendiX B

3-A Plotting program of Spatial Surfaces

a&RUN•/R SPSANoJSC2GL#SOLAR*39 200 3 99999 9 193 /656 0SJ4i8ASG vA SPS.iFORvIS MAIN

PARAMETER NP=166tNM=2OtNC --272tNPLOT=89NMi=109NPLOT2=16REAL RXINP)•RY(NP)rRZINP)tY(NM)PYEOUINPLOT2tfuM2)REAL UX4 NPI#UY(NP)rUZ(NPItTIINPoNMI #T24NPtNM1•T34NP@tJMIINTEGER SEO(NC)#4EO1(NP$

CC NP=NUMBER OF NODESC NM=NUMBER OF MOUESC NC_NUi16ER OF PLOT COMMANDSC NPLOT =NLIMBER OF PLOTSC NM2=NM/2C NPLOT2=NPLOT+2

NM21= NM2 ♦1OR=3.1416/180.0CALL PLOTSCALL PLOT(6„J#w.Uv-3)CALL FACTOR(O.5$

C READ GRID00 5 1=1#4READ(5t3339ERR =5) RX( I)vRY(I)rRZ(I1

5 CONTINUE00 10 I =1 +NPREAD(5r333) SE01(I1#RX4I)9RY(I1tRZII1

333 FORMAT(I5v3F15.1)10 CONTINUEC READ PLOT COMMANU Sf OUENCE

READI5g334)(SE0(J)tJ =1vN(;)334 FORMAT(1615)C READ EIGENWECTORS

U0 20 K =.1rNM00 22 I=1.3REA000359ERR=221 T1(IvK)•T2(IvK)tT3(1•K)

22 CONTINUE335 FORMAT(8Xve12.1#2E15zt)

00 20 1=1 •NP20 REAO(5a335) T1(IeK)vT241*K)tT3(I#K)

READ 45 3361 ((YEOU(_I*J) •J_1 •NM2$ 0I_1 tNPLOT2)

33600 30TI=ItNPLOTDO 25 J=I#NM2Y(J)=YEOU(I rJ)

25 CONTINUE00 27 J=NM21•NMY(J)=YEOU(NPLOT+I*J-NM2)

27 CONTINUE

C

102H

r

Appendix 11-3-D Plotting Program of Spatial Surfaces

L

L

CALL GRSUM( Y ► RXtRY•RZtT1 tT2tT3tUXtUY tUZ)CALL 1DRAU4UX9UYtUZ+SEQtSEQ11CALL PLOT(I0.00,09-31

30 CONTINUEC

CALL PLOT(10,[email protected]#999150 CONTINUE

STOPENO

AFOR91S SU81SUBROUTINE GRSU14IY ► RX ► RYtRZtTlrT29739UXtUYtUZ)PARAMETER NM=20tNP=166REAL T1(NPtNM1tT2(NPtNMItT3(NPtNMIREAL YINM)tRXINP) ►RYINPI#RZINPI tUX (NP)tUY(NF) tUZ(NP)SCL1=0.005SCL2=2.00 5 I=1 ► NPUX(I)=0.0UY(I) =o.o

S UZ(11 =0.000 10 K=19NM00 /0 1=1 ► NPUX(I) UX(I)+ Y(K)*T1(I ► K) +SLL2UY(I)= UY(I1+ Y(KI•T2(19K)*SCL2

10 U2(L)= UZ(I )+ YIK1*T3(ItK)*SCL2GO 20 1=1 9NPUX(I1= RX(I),,, SCL1 + UX(11UY(I) RY(I)+SCLL + UYtI)

20 UZII)= RZ(I) +SCLI + UZ(I)RETURNEND

@FOR tIS SU62SUBROUTINE URAU(UX ► UY ► UZ ► SEn.SE01)PARAMETER NP=166 ► NC=272REAL UX(NP) ► UY(NP) ► UZ(NP)INTEGER SEO(NC1 ► FLAG ► SE01(NP)00 10 1=1 9NC

J=IA63(SEQ(I))00 y K=1 ► NPIF(SE01(KI.E0.J1 KK=K

5 CONTINUEFLAG =2IFISEO(II.LT.01 FLAG=3

X UX(KK)Y=UY(KK)Z=UZ(KK)CALL TRANS(X ► Y ► Z ► XP ► YP)CALL PLOT (XP ► YP ► FLAG)

10 CONTINUERETURNEND

k

103

i

APpendix H--3-U Plotting Program of Spatial Surfaces

&FORVIS SU83SUOROUYINE T"AMSiX ► V' ► 3 ► XPrVP)REAL X*Y ► I' ► XP ► YP

w+'" rHETA^30.q02.3.1414/180.0XP=fX-Yt*COS4TNETA*ORIYP= 1X•rt•slMlrNCrA+na ► •

•' RETURN

ENO&RAPL18 LIB*JPLSvLYY+PLOT%&xor84on SP S.NPTO P-4903 4102 4101 101 102 103 -et, V$ 42 $1 4042 4083 4004-4064 4063 40614001 61 62 6) 66 -0 44 43 42 41 4042 4543 4044 *005-4025 40244023 4022 4021 21 22 22 24 25 -6 5 4 3 2 t 4002 400;4004 4005 4006 .5025 5024 $023 $022 5021 1021 1022 1023 1024 1025-1045 1044 104:1042 1041 $042 $043 5044 . 3045-3064 5063 5062 5061 1061 1062 1063 1064-106 14 108:1042 1081 5082 $083 $044-5103 4102 5101 1101 110.4, 1107 1364 1064 1045 102$ 1

t -25 5 1024 1044 1063 1081 1102-1101 1082 1062 1043 1043 4 24 45 -646 44 23 J 1022 1042 1061 1081 StOl-5102 $062 $061 t341 1321 2 22 41

6) 84 -103 83 41 42 21 1 $021 5042 5062 $083 5103-5084 $063 $04;5022 4002 4021 41 61 82 102 -101 81 4061 4042 4022 4003 5023 5044 5064-$045 5024 4004 402) 4041 4062 4002 4101-410 24083 4063 4044 4024 4005 5025-40014025 4045 4065 4044 4103 -103 84 64 45 25 6-132$ 5 24 44 6!63 102 --101 42 62 43 23 14 1024 1045-1044 1044 1023 3 22 4'61 81 4101-4t02 4062 4061 0 21 2 022 tO43 1063 1084-1103 tON3 106;

1042 1021 1 4021 4042 4062 4043 4103-4044 4063 4043 4322 4007 $021 14041 1061t082 1102-1101 1081 5061 0042 $022 4603 4023 4044 4064 =4045 4024 4004 S023 504;

$062 $062 $101- 5102 5083 $063 5044 $024 4005 4025-4006 $3 25 5045 1064 7054, S902AA00 SP5,NPTEVT&AOO SP5.26AFIN NIF4

1

104 NASA-^-Carl , L A. CMId

I

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THIS DOCUMENT HAS BEEN REPRODUCED FROM …INTRODUCTION AND SUMMARY This report presents the results of Part I of a two-part study of the dynamics and control analysis of the MPTS*